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CERN-TH/97-257UB-ECM-PF 97/26

hep-th/9709180September 1997

Duality in Quantum Field Theory(and String Theory) ∗

Luis Alvarez-Gaumea and Frederic Zamorab.

a Theory Division, CERN,

1211 Geneva 23, Switzerland.

b Departament d’Estructura i Constituents de la Materia,

Facultat de Fısica, Universitat de Barcelona,

Diagonal 647, E-08028 Barcelona, Spain.

ABSTRACTThese lectures give an introduction to duality in Quantum Field Theory. We discuss the phasesof gauge theories and the implications of the electric-magnetic duality transformation to describethe mechanism of confinement. We review the exact results of N = 1 supersymmetric QCD andthe Seiberg-Witten solution of N = 2 super Yang-Mills. Some of its extensions to String Theoryare also briefly discussed.

CERN-TH/97-257UB-ECM-PF 97/26September 1997

∗Based on a lectures delivered by L. A.-G. at The Workshop on Fundamental Particles and Interactions, held in VanderbiltUniversity, and at CERN-La Plata-Santiago de Compostela School of Physics, both in May 1997.

Contents

I The duality symmetry. 2

II Dirac’s charge quantization. 2

III A charge lattice and the SL(2,Z) group. 3

IV The Higgs Phase 4A The Higgs mechanism and mass gap. . 4B Vortex tubes and flux quantization. . . 5C Magnetic monopoles and permanent

magnetic confinement. . . . . . . . . . 6

V The Georgi-Glashow model and theCoulomb phase. 6

VI The ’t Hooft-Polyakov monopoles 7A The Topological nature of the magnetic

charge. . . . . . . . . . . . . . . . . . . 7B The ’t Hooft-Polyakov ansatz. . . . . . 8C The Bogomol’nyi bound and the BPS

states. . . . . . . . . . . . . . . . . . . 8D The θ parameter and the Witten effect. 9

VII The Confining phase. 10A The Abelian projection. . . . . . . . . . 10B The nature of the gauge singularities. . 10C The phases of the Yang-Mills vacuum. . 11D Oblique confinement. . . . . . . . . . . 11

VIII The Higgs/confining phase. 11

IX Supersymmetry 12A The supersymmetry algebra and its

massless representations. . . . . . . . . 12B Superspace and superfields. . . . . . . . 12C Supersymmetric Lagrangians. . . . . . 13D R-symmetry. . . . . . . . . . . . . . . . 15

X The uses of supersymmetry. 15A Flat directions and super-Higgs mecha-

nism . . . . . . . . . . . . . . . . . . . 15B Wilsonian effective actions and holo-

morphy. . . . . . . . . . . . . . . . . . . 16

XI N = 1 SQCD. 17A Classical Lagrangian and symmetries. . 17B The classical moduli space. . . . . . . . 18

XII The vacuum structure of SQCD withNf < Nc. 19A The Afleck-Dine-Seiberg’s superpotential. 19B Massive flavors. . . . . . . . . . . . . . 19

XIII The vacuum structure of SQCD withNf = Nc. 20A A quantum modified moduli space. . . 20B Patterns of spontaneous symmetry

breaking and ’t Hooft’s anomaly match-ing conditions. . . . . . . . . . . . . . . 21

XIV The vacuum structure of SQCD withNf = Nc + 1. 22A The quantum moduli space. . . . . . . 22B S-confinement. . . . . . . . . . . . . . . 22

XV Seiberg’s duality. 23A The dual SQCD. . . . . . . . . . . . . . 23B Nc+1 < Nf ≤ 3Nc/2. An infrared free

non-Abelian Coulomb phase. . . . . . . 23C 3Nc/2 < Nf < 3Nc. An interacting

non-Abelian Coulomb phase. . . . . . . 23

XVI N = 2 supersymmetry. 24A The supersymmetry algebra and its

massless representations. . . . . . . . . 24B The central charge and massive short

representations. . . . . . . . . . . . . . 25

XVII N = 2 SU(2) super Yang-Mills theory inperturbation theory. 25A The N = 2 Lagrangian. . . . . . . . . . 25B The flat direction. . . . . . . . . . . . . 26

XVIII The low energy effective Lagrangian. 26

XIX BPS bound and duality. 27

XX Singularities in the moduli space. 28

XXI The physical interpretation of the singu-larities. 28

XXII The Seiberg-Witten solution. 29A The inputs. . . . . . . . . . . . . . . . . 29B The geometrical picture. . . . . . . . . 29C The Physical connection with N = 2

super Yang-Mills. . . . . . . . . . . . . 30

XXIII Breaking N = 2 to N = 1. Monopolecondensation and confinement. 30

XXIV Breaking N = 2 to N = 0. 31

XXV String Theory in perturbation theory. 33A The type IIA and type IIB string theories. 34B The Type I string theory. . . . . . . . . 35C The SO(32) and E8×E8 heterotic strings. 35

XXVI D-branes. 35A Dirichlet boundary conditions. . . . . . 35B BPS states with RR charges. . . . . . . 36

1

XXVII Some final comments on nonpertur-bative String Theory. 37A D-instantons and S-duality. . . . . . . . 37B An eleventh dimension. . . . . . . . . . 37

I. THE DUALITY SYMMETRY.

From a historical point of view we can say that manyof the fundamental concepts of twentieth century Physicshave Maxwell’s equations at its origin. In particular someof the symmetries that have led to our understandingof the fundamental interactions in terms of relativisticquantum field theories have their roots in the equationsdescribing electromagnetism. As we will now describe,the most basic form of the duality symmetry also appearsin the source free Maxwell equations:

∇ · (E + iB) = 0 ,

∂

∂t(E + iB) + i∇× (E + iB) = 0. (I.1)

These equations are invariant under Lorentz transforma-tions, and making all of Physics compatible with thesesymmetries led Einstein to formulate the Theory of Rel-ativity. Other important symmetries of (I.1) are confor-mal and gauge invariance, which have later played im-portant roles in our understanding of phase transitionsand critical phenomena, and in the formulation of thefundamental interactions in terms of gauge theories. Inthese lectures however we will study the implications ofyet another symmetry hidden in (I.1): duality. The sim-plest form of duality is the invariance of (I.1) under theinterchange of electric and magnetic fields:

B → E ,

E → −B . (I.2)

In fact, the vacuum Maxwell equations (I.1) admit a con-tinuous SO(2) transformation symmetry †

(E + iB) → eiφ(E + iB) . (I.3)

If we include ordinary electric sources the equations (1.1)become:

∇ · (E + iB) = q ,

∂

∂t(E + iB) + i∇× (E + iB) = je . (I.4)

In presence of matter, the duality symmetry is not valid.To keep it, magnetic sources have to be introduced:

†Notice that the duality transformations are not a symmetryof the electromagnetic action. Concerning this issue see [1].

∇ · (E + iB) = (q + ig) ,

∂

∂t(E + iB) + i∇× (E + iB) = (je + i jm) . (I.5)

Now the duality symmetry is restored if at the same timewe also rotate the electric and magnetic charges

(q + ig) → eiφ(q + ig) . (I.6)

The complete physical meaning of the duality symme-try is still not clear, but a lot of work has been dedicatedin recent years to understand the implications of this typeof symmetry. We will focus mainly on the applicationsto Quantum Field Theory. In the final sections, we willbriefly review some of the applications to String Theory,where duality make striking an profound predictions.

II. DIRAC’S CHARGE QUANTIZATION.

From the classical point of view the inclusion of mag-netic charges is not particularly problematic. Since theMaxwell equations, and the Lorentz equations of motionfor electric and magnetic charges only involve the electricand magnetic field, the classical theory can accommodateany values for the electric and magnetic charges.

However, when we try to make a consistent quantumtheory including monopoles, deep consequences are ob-tained. Dirac obtained his celebrated quantization condi-tion precisely by studying the consistency conditions fora quantum theory in the presence of electric and mag-netic charges [2]. We derive it here by the quantizationof the angular momentum, since it allows to extend it tothe case of dyons, i.e., particles that carry both electricand magnetic charges.

Consider a non-relativistic charge q in the vicinity of amagnetic monopole of strength g, situated at the origin.

The charge q experiences a force m~r = q~r × ~B, where ~B

is the monopole field given by ~B = g~r/4πr3. The changein the orbital angular momentum of the electric chargeunder the effect of this force is given by

d

dt

(m~r × ~r

)= m~r × ~r

=qg

4πr3~r ×

(~r × ~r

)=

d

dt

(qg

4π

~r

r

). (II.1)

Hence, the total conserved angular momentum of the sys-tem is

~J = ~r ×m~r − qg

4π

~r

r. (II.2)

The second term on the right hand side (henceforth de-

noted by ~Jem) is the contribution coming from the elec-tromagnetic field. This term can be directly computedby using the fact that the momentum density of an elec-

tromagnetic field is given by its Poynting vector, ~E × ~B,

2

and hence its contribution to the angular momentum isgiven by

~Jem =

∫d3x~r × ( ~E × ~B) =

g

4π

∫d3x~r ×

(~E × ~r

r3

).

In components,

J iem =g

4π

∫d3xEj∂j(x

i)

=g

4π

∫

S2

xi ~E · ~ds− g

4π

∫d3x(~∇ · ~E) xi . (II.3)

When the separation between the electric and magneticcharges is negligible compared to their distance from the

boundary S2, the contribution of the first integral to ~Jemvanishes by spherical symmetry. We are therefore leftwith

~Jem = − gq

4πr . (II.4)

Returning to equation (II.2), if we assume that orbitalangular momentum is quantized. Then it follows that

qg

4π=

1

2n , (II.5)

where n is an integer. Equation (II.5) is the Dirac’scharge quantization condition. It implies that if thereexists a magnetic monopole of charge g somewhere in theuniverse, then all electric charges are quantized in unitsof 2π/g. If we have a number of purely electric chargesqi and purely magnetic charges gj, then any pair of themwill satisfy a quantization condition:

qigj = 2πnij . (II.6)

Thus, any electric charge is an integral multiple of 2π/gj.For a given gj , let these charges have n0j as the highestcommon factor. Then, all the electric charges are mul-tiples of q0 = n0j2π/gj. Similar considerations apply tothe quantization of the magnetic charge.

Till now, we have only dealt with particles that carryeither an electric or a magnetic charge. Consider now twodyons of charges (q1, g1) and (q2, g2). For this system, we

can repeat the calculation of ~Jem by following the stepsin (II.3) where now the electromagnetic fields are split as~E = ~E1 + ~E2 and ~B = ~B1 + ~B2. The answer is easilyfound to be

~Jem = − 1

4π(q1g2 − q2g1) r (II.7)

The charge quantization condition is thus generalized to

q1g2 − q2g14π

=1

2n12 (II.8)

This is referred to as the Dirac-Schwinger-Zwanziger con-dition [3].

III. A CHARGE LATTICE AND THE SL(2,Z)GROUP.

In the previous section we derived the quantization ofthe electric charge of particles without magnetic charge,in terms of some smallest electric charge q0. For a dyon(qn, gn), this gives q0gn = 2πn. Thus, the smallest mag-netic charge the dyon can have is g0 = 2πm0/q0, withm0 a positive integer dependent on the detailed theoryconsidered. For two dyons of the same magnetic chargeg0 and electric charges q1 and q2, the quantization con-dition implies q1 − q2 = nq0, with n a multiple of m0.Therefore, although the difference of electric charges isquantized, the individual charges are still arbitrary. Itintroduces a new parameter θ that contributes to theelectric charge of any dyon with magnetic charge g0 by

q = q0

(ne +

θ

2π

). (III.1)

Observe that the parameter θ+2π gives the same electriccharges that the parameter θ by shifting ne → ne + 1.Thus, we look at the parameter θ as an angular variable.

This arbitrariness in the electric charge of dyonsthrough the θ parameter can be fixed if the theory is CPinvariant. Under a CP transformation (q, g) → (−q, g).If the theory is CP invariant, the existence of a state(q, g0) necessarily leads to the existence of (−q, g0). Ap-plying the quantization condition to this pair, we get2q = q0 × integer. This implies that q = nq0 orq = (n + 1

2 )q0. If θ 6= 0, π, the theory is not CP invari-ant. It indicates that the θ parameter is a source of CPviolation. Later on we will identify θ with the instantonangle.

One can see that the general solution of the Dirac-Schwinger-Zwanziger condition (II.8) is

q = q0

(ne +

θ

2πnm

), (III.2)

g = nmg0 , (III.3)

with ne and nm integer numbers These equations can beexpressed in terms of the complex number

q + ig = q0(ne + nmτ) , (III.4)

where

τ ≡ θ

2π+

2πim0

q20. (III.5)

Observe that this definition only includes intrinsic pa-rameters of the theory, and that the imaginary part of τis positive definite. This complex parameter will play animportant role in supersymmetric gauge theories. Thus,physical states with electric and magnetic charges (q, g)are located on a discrete two dimensional lattice withperiods q0 and q0τ , and are represented by the corre-sponding vector (nm, ne) (see fig. 1).

3

qq0

g

q0τ

FIG. 1. The charge lattice with periods q0 and q0τ . Thephysical states are located on the points of the lattice.

Notice that the lattice of charges obtained from thequantization condition breaks the classical duality sym-metry group SO(2) that rotated the electric and mag-netic charges (I.6). But another symmetry group arisesat quantum level. Given a lattice as in figure 1 we candescribe it in terms of different fundamental cells. Differ-ent choices correspond to transforming the electric andmagnetic numbers (nm, ne) by a two-by-two matrix:

(nm, ne) → (nm, ne)

(α βγ δ

)−1

, (III.6)

with α, β, γ, δ ∈ Z satisfying αδ− βγ = 1. This transfor-mation leaves invariant the Dirac-Schwinger-Zwanzigerquantiztion condition (II.8). Hence the duality transfor-mations are elements of the discrete group SL(2,Z). Itsaction on the charge lattice can be implemented by mod-ular transformations of the parameter τ

τ → ατ + β

γτ + δ. (III.7)

This transformations preserve the sign of the imaginarypart of τ , and are generated just by the action of twoelements:

T : τ → τ + 1 , (III.8)

S : τ → −1

τ. (III.9)

The effect of T is to shift θ → θ + 2π. Its action iswell understood: it just maps the charge lattice (nm, ne)to (nm, ne − nm). As physics is 2π-periodic in θ, it is asymmetry of the theory. Then, if the state (1, 0) is in thephysical spectrum, the state (1, ne), with any integer ne,is also a physical state.

The effect of S is less trivial. If we take θ = 0 justfor simplicity, the S action is q0 → g0 and sends the lat-tice vector (nm, ne) to the lattice vector (−ne, nm). Soit interchanges the electric and magnetic roles. In terms

of coupling constants, it represents the transformationτ → −1/τ , implying the exchange between the weakand strong coupling regimes. In this respect the dual-ity symmetry could provide a new source of informationon nonpertubative physics.

If we claim that the S transformation is also a sym-metry of the theory we have full SL(2,Z) symmetry. Itimplies the existence of any state (nm, ne) in the physicalspectrum, with nm and ne relatively to-prime, just fromthe knowledge that there are the physical states ±(0, 1)and ±(1, 0). There are some examples of theories ‘dual-ity invariant’, for instance the SU(2) gauge theory withN = 4 supersymmetry and the SU(2) gauge theory withN = 2 supersymmetry and four flavors [4].

A priori however there is no physical reason to imposeS-invariance, in contrast with T -invariance. The stablephysical spectrum may not be SL(2,Z) invariant. Butif the theory still admits somehow magnetic monopoles,we could apply the S-transformation as a change of vari-ables of the theory, where a magnetic state is mapped toan electric state in terms of the dual variables. It couldbe convenient for several reasons: Maybe there are somephysical phenomena where the magnetic monopoles be-come relevant degrees of freedom; this is the case for themechanism of confinement, as we will see below. Theother reason could be the difficulty in the computationof some dynamical effects in terms of the original electricvariables because of the large value of the electric cou-pling q0. The S-transformation sends q0 to 1/q0. In termsof the dual magnetic variables, the physics is weakly cou-pled.

Just by general arguments we have learned a good dealof information about the duality transformations. Nextwe have to see where such concepts appear in quantumfield theory.

IV. THE HIGGS PHASE

A. The Higgs mechanism and mass gap.

We start considering that the relevant degrees of free-dom at large distances of some theory in 3+1 dimensionsare reduced to an Abelian Higgs model:

L(φ∗, φ, Aµ) = −1

4FµνF

µν + (Dµφ)∗(Dµφ)

− λ

2(φ∗φ−M2)2 , (IV.1)

where

Fµν = ∂µAν − ∂νAµ ,

Dµφ = (∂µ + iqAµ)φ , (IV.2)

and q is the electric charge of the particle φ.

4

An important physical example of a theory describedat large distances by the effective Lagrangian (IV.1) (inits nonrelativistic approximation) is a superconductor.Sound waves of a solid material causes complicated de-viations from the ideal lattice of the material. Conduct-ing electrons interact with the quantums of those soundwaves, called phonons. For electrons near the Fermi sur-face, their interactions with the phonons create an at-tractive force. This force can be strong enough to causebound states of two electrons with opposite spin, calledCooper pairs. The lowest state is a scalar particle withcharge q = −2e, which is represented by φ in (IV.1).To understand the basic features of a superconductor weonly need to consider its relevant self-interactions andthe interaction with the electromagnetic field resultingfrom its electric charge q. This is the dynamics whichis encoded in the effective Lagrangian (IV.1). The val-ues of the parameters λ and M2 depend of the tempera-ture T , and in general contribute to increase the energyof the system. To have an stable ground state, we re-quire λ(T ) > 0 for any value of the temperature. Butthe function M2(T ) do not need to be negative for all T .In fact, when the temperature T drops below a criticalvalue Tc, the function M2(T ) becomes positive. In suchsituation, the ground state reaches its minimal energywhen the Higgs particle condenses,

|〈φ〉| = M . (IV.3)

If we make perturbation theory around this minima,

φ(x) = M + ϕ(x), (IV.4)

with vanishing external electromagnetic fields, we findthat there is a mass gap between the ground state andthe first excited levels. There are particles of spin onewith mass square

M2V = 2qM2, (IV.5)

which corresponds to the inverse of the penetration depthof static electromagnetic fields in the superconductor.There are also spin zero particles with mass square

M2H = 2λM2. (IV.6)

So perturbation theory already shows a quite differentbehavior of the Higgs theory from the Coulomb theory.There is only one real massive scalar field and the elec-tromagnetic interaction becomes short-ranged, with thephoton correlator being exponentially suppressed. This isa distinction that must survive nonperturbatively. Butup to now, the above does not yet distinguish a Higgstheory from just any non-gauge theory with massive vec-tor particles. There is yet another new phenomena inthe Higgs mode which shows the spontaneous symmetrybreaking of the U(1) gauge theory.

B. Vortex tubes and flux quantization.

We have seen that the Higgs condensation producesthe electromagnetic interactions to be short-range. Ig-noring boundary effects in the material, the electric andmagnetic fields are zero inside the superconductor. Thisphenomena is called the Meissner effect.

If we turn on an external magnetic field H0 be-yond some critical value, one finds that small regionsin the superconductor make a transition to a ‘non-superconducting’ state. Stable magnetic flux tubes areallowed along the material, with a transverse size of theorder of the inverse of the mass gap. Their magnetic fluxsatisfy a quantization rule that can be understood onlyby a combination of the spontaneous symmetry break-ing of the U(1) gauge symmetry and some topologicalarguments.

Parameterize the complex Higgs field by

φ(x) = ρ(x)eiχ(x), (IV.7)

and perform fluctuations around the configuration whichminimizes the energy. i.e., we consider that ρ(x) ≃ Malmost everywhere, but at some points ρ may be zero. Atsuch points χ needs not be well defined and therefore inall the rest of space χ could be multivalued. For instance,if we take a closed contour C around a zero of ρ(x), thenfollowing χ around C could give values that run from 0 to2πn, with n an integer number, instead of coming backto zero. These are exactly the field configurations thatproduce the quantized magnetic flux tubes [5].

Consider a two-dimensional plane, cut somewherethrough a superconducting piece of material, with polarcoordinates (r, θ) and work in the time-like A0 = 0. Tohave a finite energy per unit length static configurationwe should demand that

φ(x) →Meiχ(θ) ,

Ai(x) →const

r, (IV.8)

for r → ∞. Obviously, to keep the fields single valued,we must have

χ(2π) = χ(0) + 2πn . (IV.9)

If n 6= 0, it is clear that at some point of the two-dimensional plane we should have that the continuousfield φ vanishes. Such field configurations do not corre-spond to the ground state.

Solve the field equations with the boundary conditions(IV.8) and (IV.9) fixed, and minimize the energy. Wefind stable vortex tubes with non-trivial magnetic fluxthrough the two-dimensional plane. To see this, performa singular gauge transformation ‡

‡Singular in the sense of being not well defined in all space.

5

φ(x) → eiqΛ(x)φ(x) ,

Aµ(x) → Aµ(x) − ∂µΛ(x) , (IV.10)

with Λ = 2πnθ/q. We compute the magnetic flux in sucha gauge and we find

Φ =

∮Aµdx

µ = Λ(2π) − Λ(0) =2πn

q. (IV.11)

It is important to realize that such field configurations,called Abrikosov vortices, are stable. The vortex tubecannot break since it cannot have an end point: as themagnetic flux is quantized, we would have be able todeform continuously the singular gauge transformationΛ to zero, something obviously not possible for n 6= 0.Physically this is the statement that the magnetic fluxis conserved, a consequence of the Maxwell equations.Mathematically it means that for n 6= 0 the functionχ(θ) belongs to a nontrivial homotopy class of the fun-damental group Π1(U(1)) = Z.

The existence of these macroscopic stable objects canbe used as another characterization of the Higgs phase.They should survive beyond perturbation theory.

C. Magnetic monopoles and permanent magneticconfinement.

The magnetic flux conservation in the Abelian Higgsmodel tells us that the theory does not include magneticmonopoles. But it is remarkable that the magnetic fluxis precisely a multiple of the quantum of magnetic charge2π/q found by Dirac. If we imagine the effective gaugetheory (IV.1) enriched somehow by magnetic monopoles,they would form end points of the vortex tubes. The en-ergy per unit length, i.e., the string tension σ, of theseflux tubes is of the order of the scale of the Higgs con-densation,

σ ∼M2. (IV.12)

It implies that the total energy of a system composed of amonopole and an anti-monopole, with a convenient mag-netic flux tube attached between them, would be at leastproportional to the separation length of the monopoles.In other words: magnetic monopoles in the Higgs phaseare permanently confined.

V. THE GEORGI-GLASHOW MODEL AND THECOULOMB PHASE.

The Georgi-Glashow model is a Yang-Mills-Higgs sys-tem which contains a Higgs multiplet φa (a = 1, 2, 3)transforming as a vector in the adjoint representationof the gauge group SO(3), and the gauge fields Wµ =W aµT

a. Here, T a are the hermitian generators of SO(3)

satisfying [T a, T b] = ifabcT c. In the adjoint representa-tion, we have (T a)bc = −ifabc and, for SO(3), fabc = ǫabc.

The field strength of Wµ and the covariant derivative onφa are defined by

Gµν = ∂µWν − ∂νWµ + ie[Wµ,Wν ] ,

Dµφa = ∂µφ

a − eǫabcW bµφ

c . (V.1)

The minimal Lagrangian is then given by

L = −1

4GaµνG

aµν

+1

2DµφaDµφ

a − V (φ) , (V.2)

where,

V (φ) =λ

4

(φaφa − a2

)2. (V.3)

The equations of motion following from this Lagrangianare

(DνGµν)a = −e ǫabc φb (Dµφ)c,

DµDµφa = −λφa(φ2 − a2) . (V.4)

The gauge field strength also satisfies the Bianchi identity

Dν Gµνa = 0 . (V.5)

Let us find the vacuum configurations in this theory.Introducing non-Abelian electric and magnetic fields,G0ia = −E ia and Gija = −ǫijkBka , the energy density is

written as

θ00 =1

2

((E ia)2 + (Bia)2

+(D0φa)

2 + (Diφa)2)

+ V (φ) . (V.6)

Note that θ00 ≥ 0, and it vanishes only if

Gµνa = 0, Dµφ = 0, V (φ) = 0 . (V.7)

The first equation implies that in the vacuum, W aµ is

pure gauge and the last two equations define the Higgsvacuum. The structure of the space of vacua is deter-mined by V (φ) = 0 which solves to φa = φavac such that|φvac| = a. The space of Higgs vacua is therefore a two-sphere (S2) of radius a in field space. To formulate a per-turbation theory, we have to choose one of these vacuaand hence, break the gauge symmetry spontaneously Thepart of the symmetry which keeps this vacuum invariant,still survives and the corresponding unbroken generatoris φcvacT

c/a. The gauge boson associated with this gener-ator is Aµ = φcvacW

cµ/a and the electric charge operator

for this surviving U(1) is given by

Q = eφcvacT

c

a. (V.8)

If the group is compact, this charge is quantized. Theperturbative spectrum of the theory can be found by ex-panding φa around the chosen vacuum as

φa = φavac + φ′a .

6

A convenient choice is φcvac = δc3a. The perturbativespectrum (which becomes manifest after choosing an ap-propriate unitary gauge) consists of a massive Higgs ofspin zero with a square mass

M2H = 2λa2, (V.9)

a massless photon, corresponding to the U(1) gauge bo-son A3

µ, and two charged massive W-bosons, A1µ and A2

µ,with square mass

M2W = e2a2. (V.10)

This mass spectrum is realistic as long as we are atweak coupling, e2 ∼ λ≪ 1. At strong coupling, nonper-turbative effects could change significatively eqs. (V.9)and (V.10). But the fact that there is an unbroken sub-group of the gauge symmetry ensures that there is somemassless gauge boson, which a long range interaction.This is the characteristic of the Coulomb phase.

VI. THE ’T HOOFT-POLYAKOV MONOPOLES

Let us look for time-independent, finite energy solu-tions in the Georgi-Glashow model. Finiteness of energyrequires that as r → ∞, the energy density θ00 given by(V.6) must approach zero faster than 1/r3. This meansthat as r → ∞, our solution must go over to a Higgsvacuum defined by (V.7). In the following, we will firstassume that such a finite energy solution exists and showthat it can have a monopole charge related to its solitonnumber which is, in turn, determined by the associatedHiggs vacuum. This result is proven without having todeal with any particular solution explicitly. Next, wewill describe the ’t Hooft-Polyakov ansatz for explicitlyconstructing one such monopole solution, where we willalso comment on the existence of Dyonic solutions. Inthe last two subsections we will derive the Bogomol’nyibound and the Witten effect.

A. The Topological nature of the magnetic charge.

For convenience, in this subsection we will use the vec-tor notation for the SO(3) gauge group indices and notfor the spatial indices.

Let ~φvac denote the field ~φ in a Higgs vacuum. It thensatisfies the equations

~φvac · ~φvac = a2 ,

∂µ~φvac − e ~Wµ × ~φvac = 0 , (VI.1)

which can be solved for ~Wµ. The most general solutionis given by

~Wµ =1

ea2~φvac × ∂µ~φvac +

1

a~φvacAµ . (VI.2)

To see that this actually solves (VI.1), note that ∂µ~φvac ·~φvac = 0, so that

1

ea2(~φvac × ∂µ~φvac) × ~φvac =

1

ea2

(∂µ~φvaca

2 − ~φvac(~φvac · ∂µφvac))

=1

e∂µ~φvac . (VI.3)

The first term on the right-hand side of Eq. (VI.2) is the

particular solution, and ~φvacAµ is the general solution tothe homogeneous equation. Using this solution, we can

now compute the field strength tensor ~Gµν . The fieldstrength Fµν corresponding to the unbroken part of thegauge group can be identified as

Fµν =1

a~φvac · ~Gµν = ∂µAν − ∂νAµ

+1

a3e~φvac · (∂µ~φvac × ∂ν ~φvac) . (VI.4)

Using the equations of motion in the Higgs vacuum itfollows that

∂µFµν = 0 , ∂µ F

µν = 0 .

This confirms that Fµν is a valid U(1) field strength ten-sor. The magnetic field is given by Bi = − 1

2ǫijkFjk. Let

us now consider a static, finite energy solution and a sur-face Σ enclosing the core of the solution. We take Σ tobe far enough so that, on it, the solution is already inthe Higgs vacuum. We can now use the magnetic fieldin the Higgs vacuum to calculate the magnetic charge gΣassociated with our solution:

gΣ =

∫

Σ

Bidsi

= − 1

2ea3

∫

Σ

ǫijk ~φvac ·(∂j~φvac × ∂k~φvac

)dsi . (VI.5)

It turns out that the expression on the right hand side isa topological quantity as we explain below: Since φ2 = a;the manifold of Higgs vacua (M0) has the topology of S2.

The field ~φvac defines a map from Σ into M0. Since Σ isalso an S2, the map φvac : Σ → M0 is characterized byits homotopy group π2(S

2). In other words, φvac is char-acterized by an integer ν (the winding number) whichcounts the number of times it wraps Σ around M0. Interms of the map φvac, this integer is given by

ν =1

4πa3

∫

Σ

1

2ǫijk~φvac ·

(∂j~φvac × ∂k~φvac

)dsi . (VI.6)

Comparing this with the expression for magnetic charge,we get the important result

gΣ =−4πν

e. (VI.7)

Hence, the winding number of the soliton determines itsmonopole charge. Note that the above equation differs

7

from the Dirac quantization condition by a factor of 2.This is because the smallest electric charge which couldexist in our model is e/2 for an spinorial representationof SU(2), the universal covering group of SO(3). Then,in this model m0 = 2.

B. The ’t Hooft-Polyakov ansatz.

Now we describe an ansatz proposed by ’t Hooft [6]and Polyakov [7] for constructing a monopole solution inthe Georgi-Glashow model. For a spherically symmet-ric, parity-invariant, static solution of finite energy, theyproposed:

φa =xa

er2H(aer) ,

W ai = −ǫaij

xj

er2(1 −K(aer)) ,

W a0 = 0 . (VI.8)

For the non-trivial Higgs vacuum at r → ∞, they choseφcvac = axc/r = axc. Note that this maps an S2 atspatial infinity onto the vacuum manifold with a unitwinding number. The asymptotic behavior of the func-tions H(aer) and K(aer) are determined by the Higgsvacuum as r → ∞ and regularity at r = 0. Explicitly,defining ξ = aer, we have: as ξ → ∞, H ∼ ξ, K → 0and as ξ → 0, H ∼ ξ, (K − 1) ∼ ξ. The mass of thissolution can be parameterized as

M =4πa

ef (λ/e2) .

For this ansatz, the equations of motion reduce to twocoupled equations for K and H which have been solvedexactly only in certain limits. For r → 0, one getsH → ec1r

2 and K = 1 + ec2r2 which shows that the

fields are non-singular at r = 0. For r → ∞, weget H → ξ + c3exp(−a

√2λr) and K → c4ξexp(−ξ)

which leads to W ai ≈ −ǫaijxj/er2. Once again, defin-

ing Fij = φcGcij/a, the magnetic field turns out to

be Bi = −xi/er3. The associated monopole charge isg = −4π/e, as expected from the unit winding numberof the solution. It should be mentioned that ’t Hooft’sdefinition of the Abelian field strength tensor is slightlydifferent but, at large distances, it reduces to the formgiven above.

Note that in the above monopole solution, the presenceof the Dirac string is not obvious. To extract the Diracstring, we have to perform a singular gauge transforma-tion on this solution which rotates the non-trivial Higgsvacuum φcvac = axc into the trivial vacuum φcvac = aδc3.In the process,the gauge field develops a Dirac string sin-gularity which now serves as the source of the magneticcharge [6].

The ’t Hooft-Polyakov monopole carries one unit ofmagnetic charge and no electric charge. The Georgi-Glashow model also admits solutions which carry both

magnetic as well as electric charges. An ansatz for con-structing such a solution was proposed by Julia and Zee[8]. In this ansatz, φa andW a

i have exactly the same formas in the ’t Hooft-Polyakov ansatz, but W a

0 is no longerzero: W a

0 = xaJ(aer)/er2. This serves as the source forthe electric charge of the dyon. It turns out that the dyonelectric charge depends of a continuous parameter and,at the classical level, does not satisfy the quantizationcondition. However, semiclassical arguments show that,in CP invariant theories, and at the quantum level, thedyon electric charge is quantized as q = ne. This canbe easily understood if we recognize that a monopole isnot invariant under a gauge transformation which is, ofcourse, a symmetry of the equations of motion. To dealwith the associated zero-mode properly, the gauge degreeof freedom should be regarded as a collective coordinate.Upon quantization, this collective coordinate leads to theexistence of electrically charged states for the monopolewith discrete charges. In the presence of a CP violatingterm in the Lagrangian, the situation is more subtle aswe will discuss later. In the next subsection, we describea limit in which the equations of motion can be solved ex-actly for the ’tHooft-Polyakov and the Julia-Zee ansatz.This is the limit in which the soliton mass saturates theBogomol’nyi bound.

C. The Bogomol’nyi bound and the BPS states.

In this subsection, we derive the Bogomol’nyi bound [9]on the mass of a dyon in term of its electric and magnetic

charges which are the sources for Fµν = ~φ · ~Gµν/a. Usingthe Bianchi identity (V.5) and the first equation in (V.4),we can write the charges as

g ≡∫

S2∞

BidSi =

1

a

∫Bai (Diφ)ad3x ,

q ≡∫

S2∞

EidSi =

1

a

∫Eai (Diφ)ad3x . (VI.9)

Now, in the center of mass frame, the dyon mass is givenby

M ≡∫d3xθ00 =

∫d3x

(1

2

[(Eak )2 + (Bak)2

+ (Dkφa)2 + (D0φ

a)2]+ V (φ)

), (VI.10)

where, θµν is the energy momentum tensor. Using (VI.9)and some algebra we obtain

M =

∫d3x

(1

2

[(Eak −Dkφ

a sin θ)2

+ (Bak −Dkφa cos θ)2 + (D0φ

a)2]

+ V (φ)) + a(q sin θ + g cos θ) , (VI.11)

where θ is an arbitrary angle. Since the terms in the firstline are positive, we can write M ≥ (q sin θ + g cos θ).

8

This bound is maximized for tan θ = q/g. Thus we getthe Bogomol’nyi bound on the dyon mass as

M ≥ a√g2 + q2 . (VI.12)

For the ’t Hooft-Polyakov solution, we have q = 0, andthus, M ≥ a|g|. But |g| = 4π/e and MW = ae = aq, sothat

M ≥ a4π

e=

4π

e2MW =

4π

q2MW =

ν

αMW .

Here, α is the fine structure constant and ν = 1 or 1/4,depending on whether the electron charge is q or q/2.Since α is a small (∼ 1/137 for electromagnetism), theabove relation implies that the monopole is much heavierthan the W-bosons associated with the symmetry break-ing.

From (VI.11) it is clear that the bound is not satu-rated unless λ → 0, so that V (φ) = 0. This is theBogomol’nyi-Prasad-Sommerfield (BPS) limit of the the-ory [9,10]. Note that in this limit, φ2

vac = a2 is no longerdetermined by the theory and, therefore, has to be im-posed as a boundary condition on the Higgs field. More-over, in this limit, the Higgs scalar becomes massless.Now, to saturate the bound we set

D0φa = 0 ,

Eak = (Dkφ)a sin θ ,

Bak = (Dkφ)a cos θ , (VI.13)

where, tan θ = q/g. In the BPS limit, one can use the ’tHooft-Polyakov (or the Julia-Zee) ansatz either in (V.4),or in (VI.13) to obtain the exact monopole (or dyon)solutions [9,10]. These solutions automatically saturatethe Bogomol’nyi bound and are referred to as the BPSstates. Also, note that in the BPS limit, all the per-turbative excitations of the theory saturate this boundand, therefore, belong to the BPS spectrum. As we willsee later, BPS states appears in a very natural way intheories with N = 2 supersymmetry.

D. The θ parameter and the Witten effect.

In this section we will show that in the presence of a θ-term in the Lagrangian, the magnetic charge of a particlealways contributes to its electric charge in the way givenby formula (III.2) [11].

To study the effect of CP violation, we consider theGeorgi-Glashow model with an additional θ-term as theonly source of CP violation:

L = −1

4F aµνF

aµν +1

2(Dµφ

a)2 − λ(φ2 − a2)2

+θe2

32π2F aµν F

aµν . (VI.14)

Here, F aµν = 12ǫµνρσF aρσ. The presence of the θ-term

does not affect the equations of motion but changes the

physics since the theory is no longer CP invariant. Wewant to construct the electric charge operator in this the-ory. The theory has an SO(3) gauge symmetry but theelectric charge is associated with an unbroken U(1) whichkeeps the Higgs vacuum invariant. Hence, we define anoperator N which implements a gauge rotation around

the φ direction with gauge parameter Λa = φa/a. Thesetransformations correspond to the electric charge. UnderN , a vector va and the gauge fields Aaµ transform as

δva =1

aǫabcφbvc , δAaµ =

1

eaDµφ

a .

Clearly, φa is kept invariant. At large distances where

|φ| = a, the operator e2πiN is a 2π-rotation about φand therefore exp (2πiN) = 1. Elsewhere, the rotationangle is 2π|φ|/a. However, by Gauss’ law, if the gaugetransformation is 1 at ∞, it leaves the physical statesinvariant. Thus, it is only the large distance behavior ofthe transformation which matters and the eigenvalues ofN are quantized in integer units. Now, we use Noether’sformula to compute N :

N =

∫d3x

(δL

δ∂0AaiδAai +

δLδ∂0φa

δφa).

Since δ~φ = 0, only the gauge part (which also includesthe θ-term) contributes:

δ

δ∂0Aai

(F aµνF

aµν)

= 4F aoi = −4Eai ,

δ

δ∂0Aai

(F aµνF

aµν)

= 2ǫijkF ajk = −4Bai .

Thus,

N =1

ae

∫d3xDi

~φ · ~E i − θe

8π2a

∫d3xDi

~φ · ~Bi

=1

eQe −

θe

8π2Qm ,

where, we have used (VI.9). Here, Qe and Qm are theelectric and magnetic charge operators with eigenvaluesq and g, respectively, and N is quantized in integer units.This leads to the following formula for the electric charge:

q = ne+θe2

8π2g .

For the ’t Hooft-Polyakov monopole, n = 1, g = −4π/e,and therefore, q = e(1 − θ/2π). For a general dyonicsolution we get

g =4π

enm, q = nee+

θe

2πnm . (VI.15)

and we recover (III.2) and (III.3) for q0 = e. In thepresence of a θ-term, a magnetic monopole always carriesan electric charge which is not an integral multiple of

9

some basic unit. In section III we introduced the chargelattice of periods e and eτ . In this parameterization, theBogomol’nyi bound (VI.12) takes the form

M ≥√

2|ae(ne + nmτ)| . (VI.16)

Notice that for a BPS state, equation (VI.16) impliesthat its mass is proportional to the distance of its latticepoint from the origin.

VII. THE CONFINING PHASE.

A. The Abelian projection.

In non-Abelian gauge theories, gauge fixing is a subjectfull of interesting surprises (ghosts, phantom solitons, ...)which often obscure the physical content of the theory[12].

’t Hooft gave a qualitative program to overcome thesedifficulties and provided a scenario that explains confine-ment in a gauge theory. The idea is to perform the gaugefixing procedure in two steps. In the first one a unitarygauge is chosen for the non-Abelian degrees of freedom.It reduces the non-Abelian gauge symmetry to the maxi-mal Abelian subgroup of the gauge group. Here one getsparticle gauge singularities §. This procedure is calledthe Abelian projection [12]. In this way, the dynamicsof the Yang-Mils theory will be reduced to an Abeliangauge theory with certain additional degrees of freedom.

We need a field that transforms without derivativesunder gauge transformations. An example is a real field,X in the adjoint representation of SU(N),

X → ΩXΩ−1. (VII.1)

Such a field can always be found; take for instanceXa = Ga12. We will use the field X to implement the uni-tary gauge condition which will carry us to the Abelianprojection of the SU(N) gauge group. The gauge is fixedby requiring that X be diagonal:

X =

λ1 0

. . .

0 λN

. (VII.2)

The eigenvalues of the matrix X are gauge invariant.Generically they are all different, and the gauge condi-tion (VII.2) leaves an Abelian U(1)N−1 gauge symmetry.It corresponds to the subgroup generated by the gaugetransformations

Ω =

eiω1 0

. . .0 eiωN

,

N∑

i=1

ωi = 0 . (VII.3)

§We will discuss the physical meaning of them later on.

There is also a discrete subgroup of transformationswhich still leave X in diagonal form. It is the Weylgroup of SU(N), which corresponds to permutations ofthe eigenvalues λi. We also fix the Weyl group with theconvention λ1 > λ2 > · · ·λN .

At this stage, we have an Abelian U(1) gauge theorywith N − 1 photons, N(N − 1) charged vector particlesand some additional degrees of freedom that will appearpresently.

B. The nature of the gauge singularities.

So far we assumed that the eigenvalues λi coincidenowhere. But there are some gauge field configurationsthat produce two consecutive eigenvalues to coincide atsome spacetime points

λi = λi+1 = λ, for certain i. (VII.4)

These spacetime points are ‘singular’ points of theAbelian projection. The SU(2) gauge subgroup corre-sponding to the 2×2 block matrix with coinciding eigen-values leaves invariant the gauge-fixing condition (VII.2).

Let us consider the vicinity of such a point. Prior tothe complete gauge-fixing we may take X to be

X =

D1 0 0 00 λ+ ǫ3 ǫ1 − iǫ2 00 ǫ1 + iǫ2 λ− ǫ3 00 0 0 D2

, (VII.5)

where D1 and D2 may safely be considered to be diag-onalized because the other eigenvalues do not coincide.With respect to that SU(2) subgroup of SU(N) that cor-responds to rotations among the ith and i+ 1st compo-nents, the three fields ǫa(x) form an isovector. We maywrite the central block as

λI2 + ǫaσa, (VII.6)

where σa are the Pauli matrices.Consider static field configurations. The points of

space where the two eigenvalues coincide correspond tothe points x0 that satisfy

ǫa(x0) = 0 . (VII.7)

These three equations define a single space point, andthen the singularity is particle-like. Which is its physicalinterpretation?.

By analyticity we have that ǫa ∼ (x − x0)a, and our

gauge condition corresponds to rotating the isovector ǫa

such that

ǫ =

(00|ǫ3|

). (VII.8)

From the previous sections, we know that the zero-pointof ǫa at x0 behaves as a magnetic charge with respect

10

to the remaining U(1) ⊂ SU(2) rotations. We realizethat those gauge field configurations that produce such agauge ‘singularities’ correspond to magnetic monopoles.

The non-Abelian SU(N) gauge theory is topologicallysuch that it can be cast into a U(1)N−1 Abelian gaugetheory, which will feature not only electrically chargedparticles but also magnetic monopoles.

C. The phases of the Yang-Mills vacuum.

We can now give a qualitative description of the pos-sible phases of the Yang-Mills vacuum. It is only thedynamics which, as a function of the microscopic bareparameters, determines in which phase the Yang-Millsvacuum is actually realized.

Classically, the Yang-Mills Lagrangian is scale invari-ant. One can write down field configurations with mag-netic charge and arbitrarily low energy. But quantumcorrections are likely to violate their masslessness. Ifdynamics simply chooses to give a positive mass to themonopoles, we are in a Higgs or Coulomb phase. We mustlook for the magnetic vortex tubes to figure out if we arein a Higgs phase. It will be a signal that the ordinaryHiggs mechanism has taken place in the Abelian gaugeformulation of the Yang-Mills theory. The role of the dy-namically generated Higgs field could be done by somescalar composite operator charged respect the U(1)N−1

gauge symmetries. There is also the possibility that noHiggs phenomenon occurs at all in the Abelian sector, orthat some U(1) gauge symmetries are not spontaneouslybroken. In this case we are in the Coulomb phase, withsome massless photons, or in a mixed Coulomb-Higgsphase.

There is a third possibility however. Maybe the quan-tum corrections give a formally negative mass squared forthe monopole: a magnetically charged object condenses.We apply an ‘electric-magnetic dual transformation’ towrite an effective Lagrangian which encodes the relevantmagnetic degrees of freedom in the infrared limit. In sucheffective Lagrangian, the Higgs mechanism takes place interms of dual variables. We are in a dual Higgs phase.We have electric flux tubes with finite energy per unit oflength. There is a confining potential between electricallycharged objects, like quarks.

In 1994, Seiberg and Witten gave a quantitative proofthat such dynamical mechanism of color confinementtakes place in N = 2 super-QCD (SQCD) broken toN = 1 [13], giving a non-trivial realization of ’t Hooftscenario. When N = 2 SQCD is softly broken to N = 0the same mechanism of confinement persists [14,15].

D. Oblique confinement.

For simplicity let us consider an SU(2) gauge group.We have seen that for a non-zero CP violating parameter

θ, the physical electric charge of a particle with electric(resp. magnetic) number ne (resp. nm) is:

q = (ne +θ

2πnm)e. (VII.9)

Dyons with large electric charges may have larger self-energies contributing positively to their mass squared. Ifthe state (ne, nm) condenses at θ ≃ 0, it is likely thatthe state (ne − 1, nm) condenses at θ ≃ 2π. It suggeststhat there is a phase transition around θ ≃ π. Suchfirst order phase transitions has been observed in softlybroken N = 2 SQCD to N = 0 [16].

’t Hooft proposed a new condensation mode at θ ≃ π[12]. He imagined the possibility that a bound state of thedyons (ne, nm) and (ne−1, nm), with zero electric chargeat θ = π, could be formed. Its smaller electric chargecould favor its condensation, leading to what he calledan oblique confinement mode. These oblique modes havealso been observed in softly broken N = 2 SQCD withmatter [14,15].

VIII. THE HIGGS/CONFINING PHASE.

In the previous section we have characterized the con-fining phase as the dual of the Higgs phase, i.e., the phys-ical states are gauge singlets made by the electric degreesof freedom bound by stable electric flux tubes. A goodgauge invariant order parameter measuring such behav-ior is the Wilson loop [17]:

W (C) = Tr exp

(ig

∮

C

dxµAµ

). (VIII.1)

For SU(N) Yang-Mills in the confining phase, for con-tours C, the Wilson loop obeys the area law,

〈W (C)〉 ∼ exp(−σ · (area)), (VIII.2)

with σ the string tension of the electric flux tube.But dynamical matter fields in the fundamental rep-

resentation immediately create a problem in identify-ing the confining phase of the theory through the Wil-son loop. The criterion used for confinement in thepure gauge theory, the energy between static sources, nolonger works. Even if the energy starts increasing as thesources separate, it eventually becomes favorable to pro-duce a particle-antiparticle pair out of the vacuum. Thispair shields the gauge charge of the sources, and the en-ergy stops growing. So even in a theory that ‘looks’ veryconfining our signal fails, and the perimeter law replaces(VIII.2),

〈W (C)〉 =∼ exp(−Λ · (perimeter)) (VIII.3)

If some scalar field is in the fundamental representationof the gauge group, there is no distinction at all betweenthe confinement phases and the Higgs phase. Using the

11

scalar field in the fundamental representation one canbuild gauge invariant interpolating operators for all pos-sible physical states. As the vacuum expectation valueof the Higgs field in the fundamental representation con-tinuously changes from large values to smaller ones, thespectrum of all physical states, and all other measurablequantities, changes smoothly [18]. There is no gauge in-variant operator which can distinguish between the Higgsor confining phases. We are in a Higgs/confining phase.

In supersymmetric gauge theories, it is common tohave scalar fields in the fundamental representation ofthe gauge group, the scalar quarks. In such situation,when the theory is not in the Coulomb phase, we will seethat the theory is presented in a Higgs/confining phase.We could take the phase description which is more ap-propriate for the theory. For instance, if the theory is inthe weak coupling region, it is better to realize it in theHiggs phase; if the theory in the strong coupling region,it is better to think it in a confining phase.

IX. SUPERSYMMETRY

A. The supersymmetry algebra and its masslessrepresentations.

The N = 1 supersymmetry algebra is written as [19]

Qα, Qα = 2σµααPµ

Qα, Qβ = 0 , Qα , Qβ = 0 . (IX.1)

Here, Q and Q are the supersymmetry generators andtransform as spin 1/2 operators, α, α = 1, 2. Moreover,the supersymmetry generators commute with the mo-mentum operator Pµ and hence, with P 2. Therefore,all states in a given representation of the algebra havethe same mass. For a theory to be supersymmetric, it isnecessary that its particle content form a representationof the above algebra. The irreducible representations of(IX.1) can be obtained using Wigner’s method.

For massless states, we can always go to a frame wherePµ = E(1, 0, 0, 1). Then the supersymmetry algebra be-comes

Qα, Qα =

(0 00 4E

).

In a unitary theory the norm of a state is always posi-tive. Since Qα and Qα are conjugate to each other, andQ1, Q1 = 0, it follows thatQ1|phys >= Q1|phys >= 0.As for the other generators, it is convenient to re-scalethem as

a =1

2√EQ2 , a† =

1

2√EQ2 .

Then, the supersymmetry algebra takes the form

a, a† = 1 , a, a = 0 , a†, a† = 0 .

This is a Clifford algebra with 2 fermionic generatorsand has a 2-dimensional representation. From the pointof view of the angular momentum algebra, a is a ris-ing operator and a† is a lowering operator for the helic-ity of massless states. We choose the vacuum such thatJ3|Ωλ >= λ|Ωλ > and a|Ωλ >= 0. Then

J3(a†|Ωλ >) = (λ− 1

2)(a†|Ωλ >). (IX.2)

The irreducible representations are not necessarilyCPT invariant. Therefore, if we want to assign physicalstates to these representations, we have to supplementthem with their CPT conjugates | − λ >CPT . If a rep-resentation is CPT self-conjugate, it is left unchanged.Thus, from a Clifford vacuum with helicity λ = 1/2 weobtain the N = 1 supermultiplet:

( |1/2 >, | − 1/2 >CPT |0 >, |0 >CPT

)(IX.3)

which contains a Weyl spinor ψ and a complex scalar φ.It is called the scalar multiplet.

The other relevant representation of a renormalizablequantum field theory is the vector multiplet. It is con-structed from a Clifford vacuum with helicity λ = 1:

( |1 >, | − 1 >CPT |1/2 >, | − 1/2 >CPT

). (IX.4)

It contains a vector Aµ and a Weyl spinor λ.

B. Superspace and superfields.

To make supersymmetry linearly realized it is conve-nient to use the superspace formalism and superfields[20]. Superspace is obtained by adding four spinor de-grees of freedom θα, θα to the spacetime coordinates xµ.Under the supersymmetry transformations implemented

by the operator ξαQα + ξαQα

with transformation pa-rameters ξ and ξ, the superspace coordinates transformas

xµ → x′µ = xµ + iθσµξ − iξσµθ ,

θ → θ′ = θ + ξ ,

θ → θ′= θ + ξ . (IX.5)

These transformations can easily be obtained by thefollowing representation of the supercharges acting on(x, θ):

Qα =∂

∂θα− iσµααθ

α∂µ ,

Qα = − ∂

∂θα

+ iθασµαα ∂µ . (IX.6)

These satisfy Qα, Qα = 2iσµαα ∂µ. Moreover, using thechain rule, it is easy to see that ∂/∂xµ is invariant under

12

(IX.5) but not ∂/∂θ and ∂/∂θ. Therefore, we introducethe super-covariant derivatives

Dα =∂

∂θα+ iσµαα ∂µ ,

Dα = − ∂

∂θα− iσµααθ

α ∂µ . (IX.7)

They satisfy Dα, Dα = −2iσµαα ∂µ and anti-commute

with Q and Q.The quantum fields transform as components of a su-

perfield defined on superspace, F (x, θ, θ). Since the θ-variables are anti-commuting, the Taylor expansion ofF (x, θ, θ) in (θ, θ) is finite, indicating that the supersym-metry representations are finite dimensional. The coeffi-cients of the expansion are the component fields.

To have irreducible representations we must imposesupersymmetric invariant constraints on the superfields.The scalar multiplet (IX.3) is represented by a chiralscalar superfield, Φ, satisfying the chiral constraint

DαΦ = 0 . (IX.8)

Note that for yµ = xµ + iθσµθ, we have Dαyµ =

0, Dαθβ = 0 . Therefore, any function of (y, θ) is a

chiral superfield. It can be shown that this also is anecessary condition. Hence, any chiral superfield can beexpanded as

Φ(y, θ) = φ(y) +√

2θψ(y) + θθF (y) . (IX.9)

Here, ψ and φ are the fermionic and scalar componentsrespectively and F is an auxiliary field linear and homo-geneous. Similarly, an anti-chiral superfield is defined byDαΦ† = 0 and can be expanded as

Φ†(y†, θ) = φ†(y†) +√

2θψ(y†) + θθF †(y†) , (IX.10)

where, yµ† = xµ − iθσµθ.The vector multiplet (IX.4) is represented off-shell by

a real scalar superfield

V = V †. (IX.11)

In local quantum field theories, spin one massless par-ticles carry gauge symmetries [21]. These symmetriescommute with the supersymmetry transformations. Fora vector superfield, many of its component fields canbe gauged away using the Abelian gauge transformationV → V + Λ + Λ†, where Λ (Λ†) are chiral (anti-chiral)superfields. In the Wess-Zumino gauge [19], it becomes

V = −θσµθAµ + iθ2θλ− iθ2θλ+

1

2θ2θ

2D .

In this gauge, V 2 = 12AµA

µθ2θ2

and V 3 = 0. The Wess-Zumino gauge breaks supersymmetry, but not the gaugesymmetry of the Abelian gauge field Aµ. The Abeliansuperfield gauge field strength is defined by

Wα = −1

4D

2DαV , W α = −1

4D2DαV .

It can be verified thatWα is a chiral superfield. Since it isgauge invariant, it can be computed in the Wess-Zuminogauge,

Wα = −iλα(y) + θαD − i

2(σµσνθ)α Fµν

+ θ2(σµ∂µλ)α , (IX.12)

where, Fµν = ∂µAν − ∂νAµ.In the non-Abelian case, V belongs to the adjoint

representation of the gauge group: V = VATA, where,

TA† = TA. The gauge transformations are now imple-mented by

e−2V → e−iΛ†

e−2V eiΛ ,

where Λ = ΛATA is a chiral superfield. The non-Abelian

gauge field strength is defined by

Wα =1

8D

2e2VDαe

−2V

and transforms as

Wα →W ′α = e−iΛWαe

iΛ .

In components, in the WZ gauge it takes the form

W aα = −iλaα + θαD

a − i

2(σµσνθ)αF

aµν

+ θ2σµDµλa, (IX.13)

where,

F aµν = ∂µAaν − ∂νA

aµ + fabcAbµA

cν ,

Dµλa

= ∂µλa

+ fabcAbµλc.

Now we are ready to construct supersymmetric La-grangians in terms of superfields.

C. Supersymmetric Lagrangians.

Clearly, any function of superfields is, by itself, a super-field. Under supersymmetry, the superfield transformsas δF = (ξQ + ξQ)F , from which the transformationof the component fields can be obtained. Note that the

coefficient of the θ2θ2

component is the field componentof highest dimension in the multiplet. Then, its vari-ation under supersymmetry is always a total derivativeof other components. Thus, ignoring surface terms, thespacetime integral of this component is invariant undersupersymmetry. This tells us that a supersymmetric La-grangian density may be constructed as the highest di-mension component of an appropriate superfield.

Let us first consider the product of a chiral and ananti-chiral superfield Φ†Φ. This is a general superfieldand its highest component can be computed using (IX.9)as

13

Φ†Φ |θ2θ

2 = − 1

4φ†2φ− 1

42φ†φ+

1

2∂µφ

†∂µφ

− i

2ψσµ∂µψ +

i

2∂µψσ

µψ + F †F . (IX.14)

Dropping some total derivatives we get the free field La-grangian for a massless scalar and a massless fermionwith an auxiliary field.

The product of chiral superfields is a chiral superfield.In general, any arbitrary function of chiral superfields isa chiral superfield:

W(Φi) = W(φi +√

2θψi + θθFi)

= W(φi) +∂W∂φi

√2θψi

+ θθ

(∂W∂φi

Fi −1

2

∂2W∂φiφj

ψiψj

). (IX.15)

W is referred to as the superpotential. Moreover, thespace of the chiral fields Φ may have a non-trivial met-ric gij in which case the scalar kinetic term, for exam-

ple, takes the form gij∂µφ†i∂µφj , with appropriate mod-

ifications for other terms. In such cases, the free fieldLagrangian above has to be replaced by a non-linear σ-model [22]. Thus, the most general N = 1 supersymmet-ric Lagrangian for the scalar multiplet is given by

L =

∫d4θK(Φ,Φ†) +

∫d2θW(Φ) +

∫d2θW(Φ†) .

Note that the θ-integrals pick up the highest componentof the superfield and in our conventions,

∫d2θ θ2 = 1

and∫d2θ θ

2= 1. In terms of the non-holomorphic

function K(φ, φ†), the metric in field space is given by

gij = ∂2K/∂φi∂φ†j , i.e., the target space for chiral su-

perfields is always a Kahler space. For this reason, thefunction K(Φ,Φ†) is referred to as the Kahler potential.

Remember that the super-field strength Wα is a chiralsuperfield spinor. Using the normalization Tr(T aT b) =12δab, we have that

Tr(WαWα) |θθ = −iλaσµDµλa

+1

2DaDa

− 1

4F aµνF aµν +

i

8ǫµνρσF aµνF

aρσ . (IX.16)

The first three terms are real and the last one is pureimaginary. It means that we can include the gauge cou-pling constant and the θ parameter in the Lagrangian ina compact form

L =1

4πIm

(τ Tr

∫d2θWαWα

)

= − 1

4g2F aµνF

aµν +θ

32π2F aµν F

aµν

+1

g2(1

2DaDa − iλaσµDµλ

a) , (IX.17)

where, τ = θ/2π + 4πi/g2.We now include matter fields by the introduction of the

chiral superfield Φ in a given representation of the gaugegroup in which the generators are the matrices T aij . The

kinetic energy term Φ†Φ is invariant under global gaugetransformations Φ′ = e−iΛΦ. In the local case, to insurethat Φ′ remains a chiral superfield, Λ has to be a chiralsuperfield. The supersymmetric gauge invariant kineticenergy term is then given by Φ†e−2VΦ. We are now ina position to write down the full N=1 supersymmetricgauge invariant Lagrangian as

L =1

8πIm

(τTr

∫d2θWαWα

)

+

∫d2θd2θ (Φ†e−2VΦ) +

∫d2θW +

∫d2θW . (IX.18)

Note that since each term is separately invariant, therelative normalization between the scalar part and theYang-Mills part is not fixed by N = 1 supersymmetry.In fact, under loop effects, by virtue of the perturbativenon-renormalization theorem [23], only the term withthe complete superspace integral

∫d2θd2θ gets an overall

renormalization factor Z(µ, g(µ)), with µ the renormal-ization scale and g(µ) the renormalized gauge couplingconstant. Observe the unique dependence on Re(τ) inZ, breaking the holomorphic τ -dependence of the La-grangian L. But quantities as the superpotential W arerenormalization group invariant under perturbation the-ory [23] (we will see dynamically generated superpoten-tials by nonperturbative effects).

In terms of component fields, the Lagrangian (IX.18)becomes

L = − 1

4g2F aµνF

aµν +θ

32π2F aµν F

aµν

− i

g2λaσµDµλ

a+

1

2g2DaDa

+ (∂µφ− iAaµTaφ)†(∂µφ− iAaµT aφ) −Daφ†T aφ

− i ψσµ(∂µψ − iAaµTaψ) + F †F

+

(−i

√2φ†T aλaψ +

∂W∂φ

F − 1

2

∂2W∂φ∂φ

ψψ + h.c.

).

(IX.19)

Here, W denotes the scalar component of the superpo-tential. The auxiliary fields F and Da can be eliminatedby using their equations of motion:

F =∂W∂φ

, (IX.20)

Da = g2(φ†T aφ) . (IX.21)

The terms involving these fields, thus, give rise to thescalar potential

V = |F |2 +1

2g2DaDa . (IX.22)

14

Using the supersymmetry algebra (IX.1) it is not difficultto see that the hamiltonian P 0 = H is a positive semi-definite operator, 〈H〉 ≥ 0, and that the ground state haszero energy if and only if it is supersymmetric invariant.At the level of local fields, the equation (IX.22) meansthat the supersymmetric ground state configuration issuch that

F = Da = 0 . (IX.23)

D. R-symmetry.

The supercharges Qα and Qα are complex spinors. Inthe supersymmetry algebra (IX.1) there is a U(1) sym-metry associated to the phase of the supercharges:

Q→ Q′ = eiβQ

Q→ Q′= e−iβQ. (IX.24)

This symmetry is called the R-symmetry. It plays an im-portant role in the study of supersymmetric gauge theo-ries.

In terms of superspace, the R-symmetry is introducedthrough the superfield generator (θQ + θQ). Then, itrotates the phase of the superspace components θ andθ in the opposite way as Q and Q. It gives different R-charges for the component fields of a superfield. Considerthat the chiral superfield Φ has R-charge n,

Φ(x, θ) → Φ′(x, θ) = einβΦ(x, e−iβθ) . (IX.25)

In terms of its component fields we have that:

φ → φ′ = einβφ ,ψ → ψ′ = ei(n−1)βψ ,F → F ′ = ei(n−2)βF .

Since d2(e−iβθ) = e2iβd2θ, we derive that the superpo-tential has R-charge two,

W(Φ) → W(Φ′, θ) = e2iβW(Φ, e−iβθ) , (IX.26)

and that the Kahler potential is R-neutral.

X. THE USES OF SUPERSYMMETRY.

A. Flat directions and super-Higgs mechanism

We have seen that the fields configuration of the super-symmetric ground state are those corresponding to zeroenergy. To find them we solve (IX.23). Consider a super-symmetric gauge theory with gauge group G, and mattersuperfields Φi in the representation R(f) of G. The clas-sical equations of motion of the Da (a = 1, ...,dimG)auxiliary fields give

Da =∑

f

φ†fTaf φf . (X.1)

The solutions ofDa = 0 usually lead to the concept of flatdirections. They play an important role in the analysisof SUSY theories. These flat directions may be lifted byF -terms in the Lagrangian, as for instance mass terms.

As an illustrative example of flat directions and someof its consequences, consider the SU(2) gauge group,one chiral superfield Q in the fundamental representa-tion of SU(2) and another chiral superfield Q in theanti-fundamental representation of SU(2). This is su-persymmetric QCD (SQCD) with one massless flavor. Inthis particular case, the equation (X.1) becomes

Da = q†σaq − qσaq†. (X.2)

The equations Da = 0 have the general solution (up togauge and global symmetry transformations)

q = q† =

(a0

), a arbitrary . (X.3)

The scalar superpartners of the fermionic quarks, (q, q),called squarks, play the role of Higgs fields. As these arein the fundamental representation of the gauge group,SU(2) is completely broken by the super-Higgs mecha-nism (for a 6= 0). It is just the supersymmetric gen-eralization of the familiar Higgs mechanism: three realscalars are eaten by the gluon, in the adjoint represen-tation, and three Weyl spinor combinations of the quarkspinors are eaten by the gluino to form a massive Diracspinor in the adjoint of SU(2). Gluons and gluinos ac-quire the classical square mass

M2g = 2g2

0|a|2, (X.4)

where g0 is the bare gauge coupling. We see that the the-ory is in the Higgs/confining phase. But there is not massgap; it remains a massless superfield. Its correspondingmassless scalar must move along some flat direction ofthe classical potential. This flat direction is given bythe arbitrary value of the real number |a|. This degen-eracy is not unphysical, as in the spontaneous breakingof a symmetry. When we move along the supersymmet-ric flat direction the physical observables change, as forinstance the gluon mass (X.4). Different values of |a| cor-respond to physically inequivalent vacua. The space theyexpand is called the moduli space. It would be nice tohave a gauge invariant parameterization of such an addi-tional parameter of the gauge invariant vacuum. It canonly come from the vacuum expectation value of somegauge invariant operator, since it is an independent newclassical parameter which does not appear in the bareLagrangian. The simplest choice is to take the followinggauge invariant chiral superfield:

M = QQ . (X.5)

Classically, its vacuum expectation value is

15

〈M〉 = |a|2, (X.6)

a gauge invariant statement and a good parameterizationof the flat direction.

There is one consequence of the flat directions in su-persymmetric gauge theories that, when combined withthe property of holomorphy, will be important to obtainexact results in supersymmetric theories. SQCD dependsof the complex coupling τ(µ) = θ(µ)/2π + 4πi/g2(µ) atscale µ. The angle θ(µ) measures the strength of CP vi-olation at scale µ. By asymptotic freedom, the theoryis weakly coupled at scales higher than the dynamicallygenerated scale |Λ|, which is defined by

Λ ≡ µ0e2πiτ(µ0)

b0 , (X.7)

where µ0 is the ultraviolet cut-off where the bare parame-ter τ0 = τ(µ0) is defined, and b0 is the one-loop coefficientof the beta function,

µ∂g

∂µ(µ) = g

(−b0(g2/16π2) + O(g4)

). (X.8)

The complex parameter Λ is renormalization group in-variant in the scheme of the Wilsonian effective actions,where holomorphy is not lost (see below). Observe alsothat the bare instanton angle θ0 plays the role of thecomplex phase of Λb0 .

At scales µ ≤ Mg all the gluons decouple and the rel-evant degrees of freedom are those of the ‘meson’ M . Itsself-interactions are completely determined by the ‘mi-croscopic’ degrees of freedom of the super-gluons andsuper-quarks. We must perform a matching conditionfor the physics at some scale of order Mg; the renormal-ization group will secure the physical equivalence at theother energies. If Mg ≫ Λ, this matching takes place atweak coupling, where perturbation theory in the gaugecoupling g is reliable, and we can trust the semiclassi-cal arguments, like those leading to formulae (X.4) and(X.6).

So far we have shown the existence of a flat directionat the classical level. When quantum corrections are in-cluded, the flat direction may disappear and a definitevalue of 〈M〉 is selected. For the Wilsonian effectivedescription in terms of the relevant degrees of freedomM , this is only possible if a superpotential W(M) is dy-namically generated for M . By the perturbative non-renormalization theorem, this superpotential can onlybe generated by nonperturbative effects, since classicallythere was no superpotential for the massless gauge singletM because of the masslessness of the quark multiplet.

If we turn on a bare mass for the quarks, m, the flatdirection is lifted at classical level and a determined valueof mass dependent function 〈M〉 is selected. But theadvantage of the flat direction to carry 〈M〉 → ∞ to beat weak coupling is not completely lost. This limit cannow be performed by sending the free parameter m tothe appropriate limit, as far as we are able to know themass dependence of the vacuum expectation value of themeson superfield M . Here holomorphy is very relevant.

B. Wilsonian effective actions and holomorphy.

The concept of Wilsonian effective action is simple.Any physical process has a typical scale. The idea of theWilsonian effective action is to give the Lagrangian ofsome physical processes at its corresponding characteris-tic scale µ:

L(µ)(x) =∑

i

gi(µ)Oi(x, µ) . (X.9)

Oi(x, µ) are some relevant local composite operators ofthe effective fields ϕa(p, µ). These are the effective de-grees of freedom at scale µ, with momentum modes prunning from zero to µ. There could be some symme-tries in the operators Oi that our physical system couldrealize in some way, broken or unbroken. The constantsgi(µ) measure the strength of the interaction Oi of ϕa atscale µ.

Behind some macroscopic physical processes, there isusually a microscopic theory, with a bare LagrangianL(µ0)(x) defined at scale µ0. The microscopic theory hasalso its characteristic scale µ0, much higher than the lowenergy scale µ. Also its corresponding microscopic de-grees of freedom, φj(p, µ0), may be completely differentthan the macroscopic ones ϕa(p, µ). The bare Lagrangianencodes the dynamics at scales below the ultraviolet cut-off µ0. The effective Lagrangian (X.9) is completely de-termined by the microscopical Lagrangian L(µ0)(x). Itis obtained by integrating out the momentum modes pbetween µ and µ0. It gives the values of the effectivecouplings in terms of the bare couplings gi0(µ0),

gi(µ) = gi(µ;µ0, gi0(µ0)) . (X.10)

In the macroscopic theory there is no reference to thescale µ0. Physics is independent of the ultraviolet cut-offµ0:

∂gi

∂µ0= 0 . (X.11)

The µ0-dependence on the bare couplings gi0(µ0) cancelthe explicit µ0-dependence in (X.10). This is the actionof the renormalization group. It allows to perform thecontinuum limit µ0 → ∞ without changing the low en-ergy physics.

In supersymmetric theories, there are some operatorsOi(z), depending only on z = (x, θ), the chiral super-space coordinate, not on θ. Clearly, their field con-tent can only be made of chiral superfields. Thoseof most relevant physical importance are the superpo-tential W(Φi, τ0,mf ), and the gauge kinetic operatorτ(µ/µ0, τ0)W

αWα. We say that the superpotential Wand the effective gauge coupling τ are holomorphic func-tions, with the chiral superfields Φi, the dimensionlessquotient µ/µ0 and the bare parameters τ0 and mf play-ing the role of the complex variables. The Kahler poten-tial K(Φ†,Φ) is a real function of the variables Φi, but

16

as far as supersymmetry is not broken and the theoryis not on some Coulomb phase, the vacuum structure isdetermined by the superpotential in the limit µ→ 0.

We know that complex analysis is substantially morepowerful than real analysis. For instance, there are a lotof real functions f(x) that at x → 0 and x → ∞ go likef(x) → x. But there is only one holomorphic functionf(z) (∂zf(z) = 0) with those properties: f(z) = z. Theholomorphic constraint is so strong that sometimes thesymmetries of the theory, together with some consistencyconditions, are enough to determine the unique possibleform of the functions W and τ [24].

An illustrative example is the saturation at one-loopof the holomorphic gauge coupling τ(µ/µ0, τ0) at any or-der of perturbation theory. Since τ0 = θ0/2π + i4π/g2

0,physical periodicity in θ0 implies

τ(µ

µ0, τ0) = τ0 +

∞∑

n=0

cn

(µ

µ0

)e2πniτ0 , (X.12)

where the sum is restricted to n ≥ 0 to ensure a welldefined weak coupling limit g0 → 0. The unique termcompatible with perturbation theory is the n = 0 term.Terms with n > 0 corresponds to instanton contributions.The function c0(t) must satisfy c0(t1t2) = c0(t1) + c0(t2)and hence it must be a logarithm. Hence

τpert

(µ

µ0, τ0

)= τ0 +

ib02π

lnµ

µ0, (X.13)

with b0 the one-loop coefficient of the beta function. Wecan use the definition (X.7) of the dynamically generatedscale Λ to absorb the bare coupling constant inside thelogarithm

τpert

(µΛ

)=ib02π

lnµ

Λ, (X.14)

showing explicitly the independence of the effective gaugecoupling in the ultraviolet cut-off µ0.

We would like to comment that the one-loop saturationof the perturbative beta function and the renormalizationgroup invariance of the scale Λ can be lost by the effectof the Konishi anomaly [25,26]. In general, after the in-tegration of the modes µ < p < µ0 the kinetic terms ofthe matter fields Φi are not canonically normalized,

L(µ) =∑

i

Zi(µ

µ0, g0)

∫d4θΦ†

ie−2VΦi + · · · (X.15)

These terms have an integral on the whole super-space (θ, θ) and hence are not protected by any non-renormalization theorem. For N = 1 gauge theories,holomorphy is absent there, and the functions Zi(

µµ0, g0)

are just real functions with perturbative multi-loops con-tributions. A canonical normalization of the matterfields in the effective action, defining the canonical fields

Φ′i = Z

1/2i Φi do not leaves invariant the path inte-

gral measure ΠiDΦi. The anomaly is proportional to

(∑

i lnZi) WαWα, giving a non-holomorphic contribu-

tion to the effective coupling τ . For N = 2 theories,Zi = 1 and holomorphy is not lost for τ [26,27].

XI. N = 1 SQCD.

A. Classical Lagrangian and symmetries.

We now analyze N = 1 SQCD with gauge groupSU(Nc) and Nf flavors ∗∗ . The field content is thefollowing: There is a spinor chiral superfield Wα in theadjoint of SU(Nc), which contains the gluons Aµ and thegluinos λ. The matter content is given by 2Nf scalar chi-

ral superfields Qf and Qf , f, f = 1, ..., Nf , in the Nc and

Nc representations of SU(Nc) respectively. The renor-malizable bare Lagrangian is the following:

LSQCD =1

8πIm

(τ0

∫d2θ WαWα

)

+

∫d4θ

(Q†fe

−2VQf + Qfe2V Q†

f

)

+

(∫d2θ mf QfQf + h.c.

), (XI.1)

with τ0 = θ0/2π+ i4π/g20 and mf the bare couplings. In

the massless limit the global symmetry of the classicalLagrangian is SU(Nf )L × SU(Nf)R ×U(1)B ×U(1)A ×U(1)R. For Nc = 2 the representations 2 and 2 are equiv-alent, and the global symmetry group is enlarged. Ingeneral we consider Nc > 2. The U(1)A and U(1)R sym-metries are anomalous and are broken by instanton ef-fects. But we can perform a linear combination of U(1)Aand U(1)R, call it U(1)AF , that is anomaly free. Wehave the following table of representations for the globalsymmetries of SQCD:

SU(Nf )L SU(Nf)R U(1)B U(1)AF

Wα 1 1 0 1

Qf Nc 1 1(Nf−Nc)

Nf

Qf 1 Nc −1(Nf−Nc)

Nf

The anomaly free R-charges, RAF , are derived by thefollowing. The superfield Wα is neutral under U(1)A andits R-transformation is fixed to be

Wα(x, θ) → eiβWα(x, e−iβθ). (XI.2)

∗∗Some reviews on exacts results in N = 1 supersymmetricgauge theories are [28].

17

Consider now that the fermionic quarks ψ have chargeRψunder an U(1)AF transformation. In the one-instantonsector, λ has 2Nc zero modes, and one for each Qf and

Qf . In total we have 2Nc + 2NfRψ = 0 to avoid theanomalies. We derive that Rψ = −Nc/Nf . Since this is

the charge of the fermions, the superfields (Qf , Qf) haveRAF charge 1 −Nc/Nf = (Nf −Nc)/Nf .

B. The classical moduli space.

The classical equations of motion of the auxiliary fieldsare

F qf= −mf qf = 0 ,

F qf= −mfqf = 0 ,

Da =∑

f

(q†fT

aqf − qfTaq†f

)= 0 . (XI.3)

If there is a massive flavor mf 6= 0, then we must haveqf = qf = 0. As we want to go to the infrared limit toanalyze the vacuum structure, the interesting case is thesituation ofNf massless flavors. If some quark has a non-zero mass m, its physical effects can be decoupled at verylow energy, by taking into account the appropriate phys-ical matching conditions at the decoupling scale m (seebelow). If all quarks are massive, in the infrared limitwe only have a pure SU(Nc) supersymmetric gauge the-ory. The Witten index of pure SU(Nc) super Yang-Millsis tr(−1)F = Nc [29]. We know that supersymmetry isnot broken dynamically in this theory, and that there areNc equivalent vacua. The 2Nc gaugino zero modes breakthe U(1)R symmetry to Z2Nc

by the instantons. ThoseNc vacua corresponds to the spontaneously broken dis-crete symmetry Z2Nc

to Z2 by the gaugino condensate〈λλ〉 6= 0.

If there are some massless super-quarks, they canhave non-trivial physical effects on the vacuum structure.Consider that we have Nf massless flavors. We can lookat the qf and qf scalar quarks as Nc ×Nf matrices. Us-ing SU(Nc)×SU(Nf ) transformations, the qf matrix canbe rotated into a simple form. There are two cases to bedistinguished:

a) Nf < Nc:In this case we have that the general solution of the

classical vacuum equations (XI.3) is:

qf = q†f =

v1 0 · · · 00 v2

. . .

0 · · · vNf

......

0 · · · 0

, (XI.4)

with vf arbitrary. These scalar quark’s vacuum expec-tation values break spontaneously the gauge group to

SU(Nc−Nf ). By the super-Higgs mechanism, N2c−(Nc−

Nf )2 = 2NcNf − N2

f chiral superfields are eaten by the

vector superfields. This leaves 2NfNc− (2NfNc−N2f ) =

N2f chiral superfields. They can be described by the me-

son operators

Mfg ≡ QfQg. (XI.5)

which provide a gauge invariant description of the clas-sical moduli space.

b) Nf ≥ Nc:In this case the general solution of (XI.3) is:

qf =

v1 0 · · · 0 · · · 0

0 v2...

.... . .

...vNc

· · · 0

, (XI.6)

q†f =

v1 0 · · · 0 · · · 0

0 v2...

.... . .

...vNc

· · · 0

, (XI.7)

with the parameters vi, vi (i = 1, ..., Nc) subject to theconstraint

|vi|2 − |vi|2 = constant independent of i. (XI.8)

Now the gauge group is completely higgsed. The gaugeinvariant parameterization of the classical moduli spacemust be done by 2NfNc−(N2

c −1) chiral superfields. Forinstance, if Nf = Nc, we need N2

c + 1 superfields. Themeson operators Mfg provide N2

c . The remaining degreeof freedom comes from the baryon-like operators

B = ǫf1···fNf Qf1 · · ·QfNf,

B = ǫf1···fNf Qf1 · · · QfNf, (XI.9)

with the color indices also contracted by the ǫ-tensor.These are two superfields, but there is a holomorphicconstraint

detM − BB = 0 . (XI.10)

For Nf = Nc + 1, we need 2Nc(Nc + 1) − (N2c − 1) =

N2c + 2Nc + 1 independent chiral superfields. We can

construct the baryon operators:

Bf = ǫff1···fNcQf1 · · ·QfNc,

Bf = ǫff1···fNc Qf1 · · · QfNc. (XI.11)

Mfg, Bf and Bf have (Nc+1)2 +2(Nc+1) components.

The matrix Mfg has rank Nc, which can be expressed bythe 2(Nc + 1) constraints:

MfgBg = MfgB

g = 0 . (XI.12)

And in total we get the needed N2c +2Nc+1 independent

chiral superfields.As Nf increases, we get more and more constraints.

Each case with Nf ≥ Nc is interesting by itself and wewill have to look at them in different ways.

18

XII. THE VACUUM STRUCTURE OF SQCDWITH NF < NC .

A. The Afleck-Dine-Seiberg’s superpotential.

First we consider the case of massless flavors. Atthe classical level there are flat directions parameter-ized by the free vacuum expectation values of the me-son fields Mfg. They belong to the representation

(Nf ,Nf , 0, 2(Nf−Nc)/Nf ) of the global symmetry groupSU(Nf)L×SU(Nf)R×U(1)B×U(1)AF . If nonperturba-tive effects generate a Wilsonian effective superpotentialW , it must depend in a holomorphic way of the light chi-ral superfields Mfg and the bare coupling constant τ0.The renormalization group invariance of the Wilsonianeffective action demands that the dependence on the barecoupling constant τ0 of W enters thought the dynami-cally generated scale ΛNf ,Nc

. The invariance of W underSU(Nf)L × SU(Nf )R rotations reduces the dependencein the mesons fields to the combination detM . There isonly one holomorphic function W = W(detM,ΛNf ,Nc

),with RAF charge two that can be built from the variablesdetM and ΛNf ,Nc

, which have RAF charge 2(Nf − Nc)and zero, respectively. It is the Afleck-Dine-Seiberg’s su-perpotential [30,31]

W = cNf ,Ng

(ΛNf ,Nc

detM

) 1(Nc−Nf )

, (XII.1)

where cNf ,Ncare some undetermined dimensionless con-

stants. If cNf ,Nc6= 0, (XII.1) corresponds to an exact

nonperturbative dynamically generated Wilsonian super-potential. It has catastrophic consequences, the the-ory has no vacuum. If we try to minimize the energyderived from the superpotential (XII.1) we find that|〈detM〉| → ∞.

B. Massive flavors.

When we add mass terms for all the flavors we expectto find some physical vacua. In fact, by Witten index,we should find Nc of them. To verify this, let us try tocompute 〈Mfg〉 taking advantage of its holomorphy andsymmetries.

A bare mass matrix mfg 6= 0 breaks explicitly theSU(Nf) × SU(Nf )R × U(1)AF global symmetry of thebare Lagrangian (XI.1). In terms of the meson operatorthe mass term is

Wtree = tr (mM). (XII.2)

We see that, under an L and R rotation of SU(Nf)L andSU(Nf)R respectively, we can recover the SU(Nf)L ×SU(Nf)R invariance if we require m to transform asm → L−1mR. In the same way, as the superpotentialhas R-charge two, the U(1)AF invariance is recovered if

we assign the charge 2 − 2(Nf − Nc)/Nf = 2Nc/Nf tothe mass matrix m. The vacuum expectation value ofthe matrix chiral superfield M is a holomorphic functionof ΛNf ,Nc

and m. To implement the same action underSU(Nf)L × SU(Nf)R rotations, we must have

〈M〉 = f(detm,ΛNf ,Nc)m−1. (XII.3)

The dependence in detm of the function f is determinedby the RAF charge. Then, the ΛNf ,Nc

dependence isworked out by dimensional analysis. The result is

〈M〉 = (const)(Λ

3Nc−Nf

Nf ,Ncdetm

) 1Ncm−1 . (XII.4)

TheNc roots giveNc vacua. Observe that this is an exactresult, and valid also for Nf ≥ Nc. There is only an di-mensionless constant (in general Nf and Nc dependent)to be determined. It would be nice to be able to carryits computation in the weak coupling limit, since holo-morphy would allow to extend (XII.4) also to the strongcoupling region.

The result (XII.4) suggest the existence of an effec-tive superpotential out of which (XII.4) can be obtained.Holomorphy and symmetries tell us that the possible su-perpotential would have to be

W(M,ΛNf ,Nc,m) =

(ΛNf ,Nc

detM

) 1(Nc−Nf )

·

f

(t = tr(mM)

(ΛNf ,Nc

detM

) −1(Nc−Nf )

). (XII.5)

In the limit of weak coupling, ΛNf ,Nc→ 0, we know that

f(t) = cNf ,Nc+ t. But we can play at the same time

with the free values of m to reach any desired value oft. This fixes the function f(t) and the superpotentialW(M,ΛNf ,Nc

,m) to be

W(M,ΛNf ,Nc,m) = cNf ,Nc

(ΛNf ,Nc

detM

) 1(Nc−Nf )

+ tr (mM). (XII.6)

As a consistency check, when we solve the equations∂W/∂M = 0, we obtain the previously determined vac-uum expectation values (XII.4).

Finally, we have to check the non-vanishing of cNf ,Nc.

We take advantage of the decoupling theorem to obtainfurther information about the constants cNf ,Nc

. Let usadd a mass term m only for the Nf flavor,

W(M,ΛNf ,Nc,m) =

(ΛNf ,Nc

detM

) 1(Nc−Nf )

+ mMNfNf. (XII.7)

Solving for the equations:

19

∂W∂MfNf

(M,ΛNf ,Nc,m) = 0,

∂W∂MNff

(M,ΛNf ,Nc,m) = 0, (XII.8)

for f < Nf gives that MfNf= MNff = 0. Hence

detM = MNfNf· detM , with M the (Nf − 1)× (Nf − 1)

matrix meson operator of the Nf −1 massless flavors. Atscales below m, the Nf -th flavor decouples and its cor-responding MNfNf

meson operator is frozen to the valuethat satisfies:

∂W∂MNfNf

(M,ΛNf ,Nc,m) = − cNf ,Nc

(Nf −Nc)·

Λ(3Nf−Nc)/(Nf−Nc)Nf ,Nc

(detM)1

(Nc−Nf )−1

detM +m = 0. (XII.9)

If we substitute the solution 〈MNfNf〉 of the previous

equation into the superpotential W(M,ΛNf ,Nc,m), we

should obtain the superpotential W(M,ΛNf−1,Nc, 0) of

Nf − 1 massless flavors with the dynamically generatedscale ΛNf−1,Nc

. The matching conditions at scale m be-tween the theory with Nf flavors and the theory withNf − 1 flavors gives the relation

mΛ3Nc−Nf

Nf ,Nc= Λ

3Nc−Nf+1Nf−1,Nc

, (XII.10)

thus,

W(M,ΛNf ,Nc,m)|〈MNf Nf

〉 = (Nc −Nf + 1) ·(

cNf ,Nc

Nc −Nf

) Nc−Nf

Nc−Nf +1(

ΛNf−1,Nc

detM

) 1(Nc−Nf +1)

, (XII.11)

and we obtain the relation

(cNf ,Nc

Nc −Nf

)Nc−Nf

=

(cNf−1,Nc

Nc −Nf + 1

)Nc−Nf+1

.

(XII.12)

Similarly, we can try to obtain another relation betweenthe constants cNf ,Nc

for different numbers of colors. Tothis end we give a large expectation value to MNfNf

with

respect the expectation values of M . Then below thescale 〈MNfNf

〉 we have SQCD with Nc − 1 colors andNf − 1 flavors. Following the same strategy as before wefind that cNf−1,Nc−1 = cNc,Nf

. It means that cNc,Nf=

cNf−Nc, which together with the relation (XII.12) gives

cNf ,Nc= (Nc −Nf )c1,2 . (XII.13)

We just have to compute the dimensionless constant c1,2of the gauge group SU(2) with one flavor. In this case,or for the general case of Nf = Nc−1, the gauge group iscompletely higgsed and there are not infrared divergencesin the instanton computation. In the weak coupling limitthe unique surviving nonperturbative contributions come

from the one-instanton sector. A direct instanton calcu-lation reveals that the constant c2,1 6= 0 [31] ††.

For Nf < Nc − 1 there is an unbroken gauge groupSU(Nc − Nf ). At scales below the smallest eigenvalueof the matrix 〈Mfg〉 we have a pure super Yang-Millstheory with Nc − Nf colors. This theory is believed toconfine with a mass gap given by the gaugino condensate〈λλ〉 6= 0. Consider the simplest case of 〈Mfg〉 = µ21Nf

.

Matching the gauge couplings at scale µ gives Λ3Nc−Nf

Nf ,Nc=

(detM) Λ3(Nc−Nf )0,Nc−Nf

, which implies for the effective super-

potential

W = (Nc −Nf)Λ30,Nc−Nf

. (XII.14)

On the other hand, the gaugino bilinear λλ is the lowestcomponent of the chiral superfield S = WαWα, whichrepresents the super-glueball operator. The bare gaugecoupling τ0 acts as the source of the operator S. If wedifferentiate (XII.14) with respect to ln Λ3(Nc−Nf ) we ob-tain the gaugino condensate

〈λλ〉 = Λ30,Nc−Nf

. (XII.15)

In fact, following the ‘integrating in’ procedure [33,34],we would obtain the Veneziano-Yankielowicz effective La-grangian [35].

It is not possible to extend the Afleck-Dine-Seiberg’ssuperpotential to the case of Nf ≥ Nc. For these valuesthe quantum corrections do not lift the flat directions,and we still have a moduli space which may be differentfrom the classical one. This is the case of Nf = Nc.

XIII. THE VACUUM STRUCTURE OF SQCDWITH NF = NC .

A. A quantum modified moduli space.

For Nf = Nc, the classical moduli space is spanned by

the gauge singlet operators Mfg, B and B subject to the

constraint detM − BB = 0. At quantum level, instantoneffects could change the classical constraint to

detM − BB = Λ2Nc , (XIII.1)

since Λ2Nc ∼ e−8π/g2+iθ corresponds to the one-instanton factor, it has the right dimensions, and theoperators (Qf , Qf ) have RAF charge zero.

To check if the quantum correction (XIII.1) really takesplace, add a mass term for the quarks. The unique pos-sible holomorphic term with RAF charge two that can begenerated with the variables (Mfg, B, B,Λ,m) is

††In the DR scheme c2,1 = 1 [32]. If we do not say thecontrary, we will work on such a scheme.

20

W = trmM . (XIII.2)

Imagine now that the Nc-flavor is much heavier, withbare mass m, than the Nc−1 other ones, with bare massmatrix m. The degree of freedom MNcNc

is given by the

constraint. Locate at B = B = MfNc= 0. By equation

(XII.4) we know that the (Nc − 1)× (Nc − 1) matrix Mis determined to be

M =(Λ2Nc+1Nc−1,Nc

detm) 1

Ncm−1, (XIII.3)

which has a non-zero determinant. It indicates that theconstraint (XIII.1) is really generated at quantum level[36]. As a final check, consider the simplest situationof Nc − 1 massless flavors. When we use the constraint(XIII.1) to express MNcNc

as function of detM we obtain

W =mΛ2Nc

detM, (XIII.4)

the Afleck-Dine-Seiberg’s superpotential for Nf = Nc−1massless flavors.

Far from the origin of the moduli field space we areat weak coupling and the quantum moduli space givenby the constraint (XIII.1) looks like the classical mod-uli space (XI.10). But far from the origin of order Λ,the one-instanton sector is sufficiently strong to changesignificatively the vacuum structure. Observe that theclassically allowed point M = B = B = 0 is not a pointof the quantum moduli space and the gluons never be-come massless.

B. Patterns of spontaneous symmetry breaking and’t Hooft’s anomaly matching conditions.

Our global symmetries are SU(Nf)L × SU(Nf )R ×U(1)B × U(1)AF . Since for Nf = Nc the super-quarksare neutral with respect to the non-anomalous symme-try U(1)AF , it is never spontaneously broken. The othersymmetries present different patterns of symmetry break-ing depending on which point of the moduli space thevacuum is located ‡‡.

For instance, the point

M = Λ21Nf, B = B = 0, (XIII.5)

suggests the spontaneous symmetry breaking

SU(Nf )L × SU(Nf )R × U(1)B × U(1)AF

−→ SU(Nf)V × U(1)B × U(1)AF , (XIII.6)

‡‡Different patterns of symmetry breaking have also beenobserved in softly broken N = 2 SQCD [15].

with SU(Nf )V the diagonal part of SU(Nf)×SU(Nf)R.To check it, the unbroken symmetries must satisfy the ’tHooft’s anomaly matching conditions [37].

With respect to the unbroken symmetries the quan-tum numbers of the elementary and composite masslessfermions, at high and low energy respectively, are

SU(Nf )V U(1)B U(1)AF

λ 1 0 1ψq Nf 1 −1ψq Nf −1 −1

ψM N2

f− 1 0 −1

ψB 1 Nf −1ψB 1 −Nf −1

Observe there are only N2f − 1 independent meson fields,

arranged in the adjoint of SU(Nf)V , since the constraint(XIII.1) eliminates one of them. There are N2

f −1 gluinosand Nf extra components for each quark ψq and anti-quark ψq because of the gauge group SU(Nc). Theanomaly coefficients are:

triangles high energy low energy

SU(Nf )2 × U(1)AF −2NfT (Nf ) −T (N2

f− 1)

U(1)3AF −2N2f + (N2

f − 1) −(N2f − 1) − 2

U(1)2B × U(1)AF −N2f −N2

f −2N2f

trU(1)AF −2N2f +N2

f − 1 −(N2f − 1) − 2

The constants T (R) are defined by tr(T aT a) =T (R)δab, with T a in the representation R of the groupSU(N). For the fundamental representation, T (N) =1/2. For the adjoint representation, T (N2 − 1) = N .The coefficient of trU(1)AF corresponds to the gravita-tional anomaly. One can check that all the anomaliesmatch perfectly, supporting the spontaneous symmetrybreaking pattern of (XIII.6).

The quantum moduli space of Nf = Nc allows anotherparticular point with a quite different breaking pattern.It is:

M = 0, B = −B = ΛNc . (XIII.7)

At this point, only the vectorial baryon symmetry is bro-ken, all the chiral symmetries SU(Nf)L × SU(Nf )R ×U(1)AF remain unbroken. We check this pattern withthe help of the ’t Hooft’s anomaly matching conditionsagain. In this case we have the quantum numbers:

21

SU(Nf)L SU(Nf)R U(1)AF

λ 1 1 1ψq Nf 1 −1ψq 1 Nf −1

ψM Nf Nf −1ψB 1 1 −1ψB 1 1 −1

and the anomaly coefficients are:

triangles high energy low energy

SU(Nf)3L NfC3 NfC3

SU(Nf )3R NfC3 NfC3

SU(Nf)2 × U(1)AF −NfT (Nf ) −NfT (Nf )

U(1)3AF −2N2f +N2

f − 1 −N2f − 1

where C3 is defined by tr(T aT b, T c) = C3dabc, with T a

in the fundamental representation of SU(Nf). Becauseof the constraint (XIII.1) there is only one independentbaryonic degree of freedom. The anomaly coefficientsmatch perfectly.

XIV. THE VACUUM STRUCTURE OF SQCDWITH NF = NC + 1.

A. The quantum moduli space.

First we consider if the classical constraints:

MfgBg = MfgB

f = 0, (XIV.1)

detM(M−1)fg − Bf Bg = 0, (XIV.2)

are modified quantum mechanically. For Nf = Nc + 1

the quark multiplets (Qf , Qf ) have RAF charge equal to1/Nf . The mass matrix breaks the U(1)AF symmetrywith a charge of 2 − 2/Nf = 2Nc/Nf . It is exactly thecharge U(1)AF of equation (XIV.2). On the other hand,the instanton factor Λ2Nc−1 supplies the right dimen-sionality. Then, there is the possibility that the classicalconstraint (XIV.2) is modified by nonperturbative con-tributions to

detM(M−1)fg −Bf Bg = Λ2Nc−1mfg. (XIV.3)

On the other hand, one can see that the classical con-straints (XIV.1) do not admit modification. Then if

M 6= 0 we have Bf = Bg = 0. Using (XII.4), we ob-tain

detM(M−1)fg = Λ2Nc−1mfg , (XIV.4)

and the quantum modification (XIV.3) really takes place[36].

B. S-confinement.

In the massless limit mfg → 0, (XIV.1) and (XIV.2)are satisfied at the quantum level. It means that the

origin of field space, M = B = B = 0, is an allowedpoint of the quantum moduli space. On such a point,there is no spontaneous symmetry breaking at all. Weuse the ’t Hooft’s anomaly matching conditions to checkit. The quantum numbers of the massless fermions athigh and low energy are:

SU(Nf)L SU(Nf )R U(1)B U(1)AF

λ 1 1 0 1ψq Nf 1 1 1

Nf− 1

ψq 1 Nf −1 1Nf

− 1

ψM Nf Nf 0 2Nf

− 1

ψB Nf 1 Nf − 1 − 1Nf

ψB 1 Nf 1 −Nf − 1Nf

and the anomaly coefficients are:

triangles high energy low energy

SU(Nf )3 NcC3 NfC3 + C3

SU(Nf )2 NcT (Nf )(−Nc

Nf) NfT (Nf )(

2Nf

− 1)

×U(1)AF +T (Nf )(− 1Nf

)

U(1)2B × U(1)AF 2NcNf (−Nc

Nf) 2NfN

2c (− 1

Nf)

U(1)3AF (N2c − 1) N2

f ( 2Nf

− 1)3

+2NfNc(−Nc

Nf)3 +2Nf(− 1

Nf)3

trU(1)AF (N2c − 1) N2

f ( 2Nf

− 1)

+2NfNc(−Nc

Nf) +2Nf(− 1

Nf)

with complete agreement. Hence, at the origin of fieldspace we have massless mesons and baryons, and thefull global symmetry is manifest. It is a singular point,with the number of massless degrees of freedom largerthan the dimensionality of the space of vacua. As wemove along the moduli space away from the origin, the‘extra’ fields become massive and the massless fluctua-tions match with the dimensionality of the moduli space.As we are in a Higgs/confining phase, there should be asmooth connection of the dynamics at the origin of fieldspace with the one away from it. This dynamics must begiven by some nonperturbative superpotential of mesonsand baryons. A theory with the previous characteristicsis called s-confining.

There is a unique effective superpotential yielding allthe constraints [36],

22

W =1

Λ2Nf−3(BgMgfB

f − detM) , (XIV.5)

it satisfies:i) Invariance under all the symmetries.ii) The equations of motion ∂W/∂M = ∂W/∂B =

∂W/∂B = 0 give the constraints (XIV.1, XIV.2).iii) At the origin all the fields are massless.

iv) Adding the bare term tr (mM) + bfBf + bf B

f werecover the Nf < Nc + 1 results.

XV. SEIBERG’S DUALITY.

A. The dual SQCD.

If we try to extend the same view of SU(Nc) SQCD forthe case ofNf > Nc+1, i.e., as being in a Higgs/confiningphase with the vacuum structure determined by mesonand baryons operators satisfying the corresponding clas-sical constraints, to the case of Nf > Nc+1 (it is not pos-sible to modify the classical constraints for Nf > Nc+1),we obtain inconsistencies. It is not possible to generatea superpotential yielding to the constraints, and the ’tHooft’s anomaly matching conditions are not satisfied.It indicates that for Nf > Nc + 1 the Higgs/confiningdescription of SQCD at large distances in terms of just

M , B and B is no longer valid.For Nf > Nc + 1, Seiberg conjectured [38] that the

infrared limit of SQCD with Nf flavors admits a dualdescription in terms of an N = 1 super Yang-Mills gauge

theory with Nc = Nf −Nc number of colors, Nf flavors

Df and Df in the fundamental and anti-fundamentalrepresentations of SU(Nf − Nc) respectively, and N2

f

gauge singlet chiral superfields M(m)gf . The fields M

(m)gf

couple to Df and Df through the relevant bare superpo-tential

W = M(m)gf DgDf . (XV.1)

If both theories are going to describe the same physicsat large distances, we must be able to give a prescrip-

tion of the gauge invariant operators Mgf , Bf1···f

Nc

and Bf1···f

Nc in terms of the dual microscopic operators

(Df , Df ) and M(m)gf . The simplest identification is:

Mgf = µM(m)gf ,

Bf1···f

Nc = Df1 · · ·DfNc ,

Bf1···f

Nc = Df1 · · · DfNc . (XV.2)

In the baryon operators the SU(Nc) color indices of

(Df , Df ) are contracted with the Nc antisymmetric ten-sor. The scale µ is introduced because the dimension

of the bare operator M(m)gf , derived from (XV.1), is one.

This mass scale relates the intrinsic scales Λ and Λ of theSU(Nc) and SU(Nc) gauge theories through the equation

Λ3Nc−Nf Λ3Nc−Nf = (−1)Nf−NcµNf . (XV.3)

We see that an strongly coupled SU(Nc) gauge theory

corresponds to a weakly coupled SU(Nc) gauge theory,in analogy with the electric-magnetic duality. From thisanalogy, we call the SU(Nc) gauge theory the electric

one, and the SU(Nc) gauge theory the magnetic one.Both theories must have the same global symmetries.

The mapping (XV.2) gives the quantum numbers of themagnetic degrees of freedom. Once more, ’t Hooft’sanomaly matching conditions for the electric and mag-netic theories give a non-trivial check of (XV.2). In thefollowing table we write the quantum numbers for thefermions of the magnetic theory:

SU(Nf )L SU(Nf )R U(1)B U(1)AF

λ 1 1 0 1

ψd Nf 1 Nc

Nc

Nc

Nf

ψd 1 Nf −Nc

Nc

Nc

Nf

ψm Nf Nf 0 1 − 2Nc

Nf

with λ the magnetic gluinos. One can check that boththeories give the same anomalies.

It can be verified that applying duality again we obtainthe original theory.

B. Nc + 1 < Nf ≤ 3Nc/2. An infrared free non-AbelianCoulomb phase.

In this range of Nf the magnetic theory is not asymp-totically free and has a trivial infrared fixed point. Atlarge distances the physical effective degrees of freedomare the fields Df , Df , Mgf and the massless super-gluonsof the gauge group SU(Nf −Nc). At the origin of fieldspace we are in an infrared free non-Abelian Coulombphase, with a complete screening of its charges in theinfrared limit. Observe that the strongly coupled elec-tric theory is weakly coupled in terms of the magneticdegrees of freedom, according to the philosophy of theelectric-magnetic duality.

C. 3Nc/2 < Nf < 3Nc. An interacting non-AbelianCoulomb phase.

As in QCD, the N = 1 SQCD has a Banks-Zaks fixedpoint [39] for Nc, Nf → ∞, when Nf/Nc = 3 − ǫ withǫ ≪ 1. We still have asymptotic freedom and under therenormalization group transformations the theory flows

23

from the ultraviolet free fixed point to an infrared fixedpoint with a non-zero finite value of the gauge couplingconstant. If there is an interacting superconformal gaugetheory the scaling dimensions of some gauge invariantoperators should be non-trivial.

The superconformal invariance includes an R-symmetry, from which the scaling dimensions of the op-erators satisfy the lower bound

D ≥ 3

2|R| (XV.4)

with equality for chiral and anti-chiral operators. TheR-current is in the same supermultiplet as the energy-momentum tensor, whose trace anomaly is zero on thefixed point. It implies that there the R-symmetry mustbe the anomaly-free U(1)AF symmetry. It gives the scal-ing dimensions of the following chiral operators:

D(M) =3

2RAF (M) = 3

Nf −NcNf

, (XV.5)

D(B) = D(B) =3

2

Nc(Nf −Nc)

Nf. (XV.6)

Unitarity restricts the scaling dimensions of the gaugeinvariant operators to be D ≥ 1. If D = 1, the cor-responding operator O satisfies the free equation of mo-tion ∂2O = 0. If D > 1, there are non-trivial interactionsbetween the operators.

For the range 3Nc/2 < Nf < 3Nc, the gauge invari-

ant chiral operators M , B and B satisfy the unitarityconstraint with D > 1. Seiberg conjectured the exis-tence of such a non-trivial fixed point for any value of3Nc/2 < Nf < 3Nc, at least for large Nc.

As 32 (Nf − Nc) < Nf < 3(Nf − Nc), there is also a

non-trivial fixed point in the magnetic theory. Seiberg’sclaim is that both theories flow to the same infrared fixedpoint [38].

XVI. N = 2 SUPERSYMMETRY.

A. The supersymmetry algebra and its masslessrepresentations.

The N = 2 supersymmetry algebra, without centralcharge, is

Q(I)α , Qβ(J) = 2(σµ)αβPµδ

IJ ,

Q(I)α , Q

(J)β = 0 (XVI.1)

with I, J = 1, 2. The algebra (XVI.1) has a new symme-try. We can perform unitary rotations of the two super-

charges Q(I)α that do leave the anti-commutator relations

(XVI.1) invariant. We have an U(2)R = U(1)R×SU(2)Rsymmetry. The Abelian factor U(1)R corresponds to the

familiar R-symmetry of supersymmetric theories that ro-

tate the global phase of the supercharges Q(I)α . With re-

spect the SU(2)R group, the superchargesQ(I)α are in the

doublet representation 2.As in massless N = 1 supersymmetric representations,

half of the supercharges are realized as vanishing opera-

tors: Q(I)2 = 0. We normalize the other two supercharges,

a(I)1 =

1

2√EQ

(I)1 , (XVI.2)

which are an SU(2)R doublet. The massless N = 2 vec-tor multiplet is a representation constructed from theClifford vacuum |1 >, which has helicity λ = 1 and isan SU(2)R singlet. From it we obtain two fermionicstates, |1/2 >(I)= (a(I))†|1 >, and a scalar boson |0 >=(a(1))†(a(2))†|1 >. After CPT doubling we obtain theN = 2 vector multiplet:

|1 >, | − 1 >CPT

| 12 >(1), | − 12 >

(1)CPT | 12 >(2), | − 1

2 >(2)CPT

|0 >, |0 >CPT

(XVI.3)

In terms of local fields we have: a vector Aµ (the gaugebosons of some gauge group G, since we consider mass-less representations), which is SU(2)R singlet; two Weylspinors λ(I), the gauginos, arranged in an SU(2)R dou-blet; and a complex scalar φ, playing the role of the Higgs,a singlet of SU(2)R but in the adjoint of the gauge groupG. These fields arrange as

Aµւ

λ(1) λ(2)

ւφ

(XVI.4)

where the arrows indicate the action of the supercharge

Q(1)

·α . We can use a manifest N = 1 supersymmetryrepresentation taking into account that the N = 2 vec-tor multiplet is composed of an N = 1 vector multi-plet Wα = (Aµ, λ

(1)) and an N = 1 chiral multiplet

Φ = (φ, λ(2)).The massless N = 2 hypermultiplet is a representation

constructed from a Clifford vacuum |1/2 >, which is anSU(2)R singlet. The action of the two grassmanian oper-ators aIα seems to produce the same particle content thatthe N = 1 chiral multiplet, but |1/2 >= |1/2,R > isusually in some non-trivial representation R of a gaugegroup G. As R → R under a CPT transformation, itforces to make the CPT doubling, and the N = 2 hy-permultiplet is built from two N = 1 chiral multiplets incomplex conjugate gauge group representations:

24

| 12 ,R > , | − 12 ,R >CPT

|0,R >(1), |0,R >(1)CPT |0,R >(2) , |0,R >

(2)CPT

| − 12 ,R > , |12 ,R >CPT

(XVI.5)

Which represents the local fields

ψqւ

q q†

ւψq

(XVI.6)

with the complex scalar fields (q, q†) in a doublet rep-resentation of SU(2)R. In terms of N = 1 superfieldswe have one chiral superfield Q = (q, ψq) in gauge repre-

sentation R and another chiral superfield Q = (q, ψq) in

gauge representation R. All the field in the hypermulti-plet have spin ≤ 1/2. Because of the CPT doubling, thematter content of extended supersymmetry (N > 1) isalways in vectorial representations of the gauge group.

B. The central charge and massive shortrepresentations.

As shown by Haag, Lapuszanski and Sohnius [40], theN = 2 supersymmetry algebra admits a central exten-sion:

Qaα, Qbβ = 2√

2ǫαβǫabZ ,

Qαa, Qβb = 2√

2ǫαβǫabZ . (XVI.7)

Since Z commutes with all the generators, we can fix itto be the eigenvalue for the given representation. Now,let us define:

aα =1

2Q1

α + ǫαβ(Q2β)

† , (XVI.8)

bα =1

2Q1

α − ǫαβ(Q2β)

† . (XVI.9)

Then, in the rest frame, the N = 2 supersymmetry alge-bra reduces to

aα, a†β = δαβ(M +√

2Z) , (XVI.10)

bα, b†β = δαβ(M−√

2Z) , (XVI.11)

with all other anti-commutators vanishing. Since allphysical states have positive definite norm, it follows thatfor massless states, the central charge is trivially realized(i.e.,,Z = 0), as we used before. For massive states,

this leads to a bound on the mass M ≥√

2|Z|. When

M =√

2|Z|, the operators in (XVI.11) are trivially real-ized and the algebra resembles the massless case. The di-mension of the representation is greatly reduced. For ex-ample, a reduced massive N = 2 multiplet has the samenumber of states as a massless N = 2 multiplet. Thusthe representations of the N = 2 algebra with a centralcharge can be classified as either long multiplets (when

M >√

2|Z|) or short multiplets (when M =√

2|Z|).From (XVI.11) it is clear that the BPS states [9,10]

(which saturate the bound) are annihilated by half of thesupersymmetry generators and thus belong to reducedrepresentations of the supersymmetry algebra. An im-portant consequence of this is that, for BPS states, therelationship between their charges and masses is dictatedby supersymmetry and does not receive perturbative ornonperturbative corrections in the quantum theory. Thisis so because a modification of this relation implies thatthe states no longer belong to a short multiplet. On theother hand, quantum corrections are not expected to gen-erate the extra degrees of freedom needed to convert ashort multiplet into a long multiplet. Since there is noother possibility, we conclude that for short multipletsthe relation M =

√2|Z| is not modified either perturba-

tively or nonperturbatively.

XVII. N = 2 SU(2) SUPER YANG-MILLS THEORYIN PERTURBATION THEORY.

A. The N = 2 Lagrangian.

The N = 2 superspace has two independent chiralspinors θ(I), I = 1, 2. The N = 2 vector multiplet can bewritten in terms of N = 2 superspace by the N = 2 su-perfield Ψ(x, θ(I)) subject to the superspace constraints[41]:

∇(I)

·α Ψ = 0 ,

∇(I)∇(J)Ψ = ǫIKǫJL∇(K)∇(L)

Ψ . (XVII.1)

where ∇(I)α = D(I)α + Γ(I)α is the generalized super-

covariant derivative of the variable θ(I), with Γ(I)α thesuperconnection. The N = 1 superfields are connectedto the N = 2 vector superfield through the equations:

Ψ|θ(2)=θ

(2)=0

= Φ(x, θ(1), θ(1)

) ,

∇(2)αΨ|θ(2)=θ

(2)=0

= i√

2Wα(x, θ(1), θ1) . (XVII.2)

It results that the renormalizable N = 2 super Yang-Mills Lagrangian is

L =1

8πIm

(τ

∫d2θ(1)d2θ(2) ΨaΨa

)(XVII.3)

with our old friend τ = θ/2π+i4π/g2. In terms of N = 1superspace, using (XVII.1) and (XVII.2), with θ ≡ θ(1),the Lagrangian is

25

L =1

8πIm

(τ

∫d2θ WαWα

)+

1

g2

∫d2θd2θ Φ†e−2VΦ .

(XVII.4)

It looks like N = 1 SU(2) gauge theory with an ad-joint chiral superfield Φ. The point is that the 1/g2

normalization in front of the kinetic term of Φ givesN = 2 supersymmetry. In fact, when we perform theremaining superspace integral in (XVII.4), we obtain aLagrangian that looks like a Georgi-Glashow model witha complex Higgs triplet and the addition of a Dirac spinor

(λ(1), λ(2)

) in the adjoint also. This Lagrangian doesnot have all the gauge invariant renormalizable terms.N = 2 supersymmetry restricts the possible terms andgives relations between their couplings, such that at theend there are only the parameters g2 and θ.

If we apply perturbation theory to the Lagrangian(XVII.3) we only have to perform a one loop renormaliza-tion. This is an indication that in N = 2 supersymmetry,holomorphy is not lost by radiative corrections. The rea-son is the following: We expained that the multi-looprenormalization of the coupling τ came from the gener-ation of non-holomorphic factors Z(µ/µ0, g) in front ofthe complete N = 1 superspace integrals. At the level ofthe Lagrangian (XVII.4), consider the bare coupling τ0at scale µ0 and integrate out the modes between µ0 andµ. If we consider only the renormalizable terms, N = 1supersymmetry gives us

Lren =1

8πIm

(τ(µ/Λ)

∫d2θWαWα

)

+ Z

(µ

µ0, g0

)1

g2(µΛ)

∫d2θd2θ Φ†e−2VΦ (XVII.5)

where

τ(µ

Λ) =

2i

πlnµ

Λ+

∞∑

n=0

cn

(Λ

µ

)4n

(XVII.6)

is the renormalized coupling constant at scale µ. We usedthe one-loop beta function of N = 2 SU(2) gauge theoryb0 = 4 and the renormalization group invariant scale Λ ≡µ0exp(iπτ0/2). The dimensionless constants cn are thecoefficients of the n-instanton contribution (Λ/µ)4n =exp(−8πn/g2(µ) + iθ(µ)n).

If we compare with the N = 2 renormalizable La-grangian (XVII.4) we derive that Z(µ/µ0, g0) = 1. Then,there is no Konishi anomaly and the one-loop renormal-ization of τ is all there is in perturbation theory.

B. The flat direction.

Unlike N = 1 super Yang-Mills, N = 2 super Yang-Mills theory includes a complex scalar φ in the adjointof the gauge group. This scalar plays the role of a Higgs

field through the potential derived from the Lagrangian(XVII.4),

V (φ, φ†) =1

2g2[φ†, φ]2 . (XVII.7)

The supersymmetric minimum is obtained by the solu-tion of

[φ†, φ] = 0 , (XVII.8)

whose solution, up to gauge transformations, is φ = aσ3,with a an arbitrary complex number. This is our flatdirection. Along it, the SU(2) gauge group is spon-taneously broken to the U(1) subgroup. The Ψ± =1√2(Ψ1 ± iΨ2) superfield components have U(1) electric

charge Qe = ±g, respectively, and they have the classicalsquared mass

M2W = 2|a|2 . (XVII.9)

The A ≡ Ψ3 superfield component remains massless. Weknow that the Lagrangian (XVII.3) admits semi-classicaldyons with electric charge Qe = neg+θ/2π and magneticcharge Qm = (4π/g), i.e., the points (1, ne) in the chargelattice. They have the classical squared mass

M2(1, ne) = 2|a|2|ne + τ |2 . (XVII.10)

Physical masses are gauge invariant. We can use thegauge invariant parametrization of the moduli space interms of the chiral superfield

U = trΦ2 , (XVII.11)

and traslate the a-dependence in previous formulae byan u-dependence through the relation u = tr〈φ2〉. Theclassical relation is just u = a2/2.

Then, semi-classical analysis gives A as the uniquelight degree of freedom. Only at u = 0 the full SU(2)gauge symmetry is restored. How is this picture modifiedby the nonperturbative corrections?. The Seiberg-Wittensolution answers this question [13] §§.

XVIII. THE LOW ENERGY EFFECTIVELAGRANGIAN.

The N = 2 vector superfield A is invariant under theunbroken U(1) gauge transformations. At a scale of theorder of the MW mass, i.e., of the order or |u|1/2, themost general N = 2 Wilsonian Lagrangian, with twoderivatives and four fermions terms, that can be con-structed from the light degrees of freedom in A is

§§Some additional reviews on the Seiberg-Witten solutionare [42].

26

Leff =1

4πIm

(∫d2θ(1)d2θ(2) F(A)

)(XVIII.1)

with F a holomorphic function of A, called the pre-potential. We stress that the unique inputs to equa-tion (XVIII.1) are N = 2 supersymmetry and that Ais a vector multiplet. We derive an immediate conse-quence of the general form of the effective Lagrangian(XVIII.1): N = 2 supersymmetry prevents the genera-tion of a superpotential for the N = 1 chiral superfieldof A. It means that the previously derived flat direction,parametrized by the arbitrary value u = tr〈φ2〉, is notlifted by nonperturbative corrections.

In terms of N = 1 superspace we have

Leff =1

4πIm

(∫d2θ

1

2τ(A)WαWα

)

+

∫d2θd2θ K(A,A) , (XVIII.2)

where

τ(A) =∂2F∂A2

(A), (XVIII.3)

K(A,A) = Im

(∂F∂A

A

), (XVIII.4)

and A is the N = 1 chiral multiplet of A.The Wilsonian Lagrangian (XVIII.2) is an Abelian

gauge theory defined at some scale of order MW ∼|u|1/2. Interaction terms come out after the expansion

A = a + A, with a the vacuum expectation value of theHiggs field, and A the quantum fluctuations of the chiralsuperfield. The matching at scale |u|1/2 with the high en-ergy SU(2) theory is performed by the renormalizationgroup:

τ(u) =i

πlnu

Λ2+

∞∑

n=0

cn

(Λ2

u

)2n

. (XVIII.5)

Observe that the phase of the dimensionless quotientu/Λ2 plays the role of the bare θ0 angle. If we are able toknow the relation between the u and a variables, i.e., thefunction u(a), we can replace it into (XVIII.5) to obtainτ(a). Integrating twice in the variable a we obtain theprepotential

F(a) =i

2πa2ln

a2

Λ2+ a2

∞∑

n=1

Fk(

Λ

a

)4k

. (XVIII.6)

If we look at the terms of the Lagrangian (XVIII.2) pro-portional to the dimensionless constant Fn, they corre-spond to the effective interaction terms created by then-instanton contribution, as expected. For a → ∞, theinstanton contributions go to zero. This is an expectedresult, since at a→ ∞ the matching takes place at weakcoupling due to asymptotic freedom. In this region per-turbation theory is applicable and we can believe thesemi-classical relation, u ∼ a2/2.

XIX. BPS BOUND AND DUALITY.

The N = 2 supersymmetry algebra gives the massbound

M ≥√

2|Z| , (XIX.1)

with Z the central charge. The origin of the centralcharge is easy to understand: the supersymmetry chargesQ and Q are space integrals of local expressions in thefields (the time component of the super-currents). Incalculating their anti-commutators, one encounters sur-face terms which are normally neglected. However, inthe presence of electric and magnetic charges, these sur-face terms are non-zero and give rise to a central charge.When one calculates the central charge that arises fromthe classical Lagrangian (XVII.3) one obtains [43]

Z = ae(n+mτ) , (XIX.2)

so that M ≥√

2|Z| coincides with the Bogomol’nyibound (VI.12).

But the equation (XIX.2) is a classical result. Theeffective Lagrangian (XVIII.1) includes all the nonper-turbative quantum corrections of the higher modes. Toget their contribution to the BPS bound, we just haveto compute the central charge that is derived from theeffective Lagrangian (XVIII.1). The result is

Z(nm, ne) = nea+ nmaD , (XIX.3)

for a supermultiplet located in the charge lattice at(nm, ne). We have defined the aD function

aD ≡ ∂F∂a

(a) . (XIX.4)

This function plays a crucial role in duality. Observe that

under the SL(2,Z) transformation M =

(α βγ δ

)of the

charge lattice,

(nm, ne) → (nm, ne)M−1 , (XIX.5)

the invariance of the central charge demands

(aDa

)→ M

(aDa

). (XIX.6)

Its action on the effective gauge coupling τ = ∂aD/∂a is

τ → ατ + β

γτ + δ. (XIX.7)

The S-transformation, that interchanges electric withmagnetic charges, makes

aD → a ,

a→ −aD . (XIX.8)

27

Then, aD is the dual scalar photon, that couples locallywith the monopole (1, 0) through the dual gauge couplingτD = −1/τ .

From (XVIII.3) and (XVIII.4), we see that Imτ(a) isthe Kahler metric of the Kahler potential K(a, a),

d2s = [Imτ(a)]dada . (XIX.9)

Physical constraints demands the metric be positive def-inite, Imτ > 0. However, if τ(a) is globally defined themetric cannot be positive definite as the harmonic func-tion Imτ(a) cannot have a minimum. This indicates thatthe above description of the metric in terms of the vari-able a must be valid only locally. In the weak couplingregion, |u| ≫ |Λ|, where τ(a) ∼ (2i/π)ln(a/Λ), we havethat Imτ(a) > 0, but for a ∼ Λ, when the theory is atstrong coupling and the nonperturbative effects becomeimportant, the perturbative result does not give the cor-rect physical answer. Two things should happen: theinstanton corrections must secure the positivity of themetric and physics must be described in terms of a newlocal variable a′. Which is this new local variable? If wedo not want to change the physics, the change of vari-ables must be an isometry of the Kahler metric (XIX.9).In terms of the variables (aD, a) the Kahler metric is

d2s = Im(daDda) = − i

2(daDda− dadaD) , (XIX.10)

The complete isometry group of (XIX.10) is

(aDa

)→

M

(aDa

)+

(pq

)with M ∈ SL(2,R) and p, q ∈ R.

But the invariance of the central charge puts p = q = 0∗∗∗ and the Dirac quantization condition restricts M ∈SL(2,Z). We arrive to an important result: in someregion of the moduli space we have to perform an electric-magnetic duality transformation.

XX. SINGULARITIES IN THE MODULI SPACE.

As Imτ cannot be globally defined on the u plane, theremust be some singularities ui indicating the multivalued-ness of τ(u). If we perform a loop arround a singularityui, there is a non-trivial monodromy action Mi on τ(u).This action should be an isometry of the Kahler metric,if we do not want to change the physics. It implies thatthe monodromies Mi are elements of the SL(2,Z) group.

In fact, we have found already one non-trivial mon-odromy because of the perturbative contributions. Themultivalued logarithmic dependence of τ gives the mon-odromy. For u ∼ ∞, τ ∼ (i/π)ln(u/Λ2). In that region,the loop u→ e2πiu applied on τ(u) gives

∗∗∗In N = 2 SQCD with massive matter, the central chargeallows to have p, q 6= 0 [44].

τ → τ − 2 . (XX.1)

Its associated monodromy is

M∞ =

(−1 20 −1

)= PT−2 . (XX.2)

which acts on the variables (aD, a) as

aD → −aD + 2a , (XX.3)

a→ −a . (XX.4)

As it should be, the monodromy is a symmetry of thetheory. T−2 just shifts the θ parameter by −4π, and Pis the action of the Weyl subgroup of the SU(2) gaugegroup. Then, the monodromy at infinity M∞ leaves thea variable invariant (up to a gauge transformation).

The monodromy at infinity means there must be somesingularity in the u plane. How many singularities?. Weknow that the anomalous U(1)R symmetry is broken byinstantons, and that there is an unbroken Z8 subgroupbecause the one-instanton sector has eight fermionic zeromodes. The U = trΦ2 operator has R-charge four. Itmeans that the u→ −u symmetry is spontaneously bro-ken, leading to equivalent physical vacua. Then, if u0 is asingular point, −u0 must be also another singular point.

Let us assume that there is only one singularity. Ifthis were the situation, the monodromy group would beAbelian, generated only by the monodromy at infinity.From the monodromy invariance of the variable a underM∞, we would have that a is a good variable to describethe physics of the whole moduli space. This is in contra-diction with the holomorphy of τ(a).

Seiberg and Witten made the assumption that thereare only two singularities, which they normalized to beu1 = Λ2 and u2 = −Λ2. This assumption leads to aunique and elegant solution that passes many tests.

XXI. THE PHYSICAL INTERPRETATION OFTHE SINGULARITIES.

The most natural physical interpretation of singulari-ties in the u plane is that some additional massless par-ticles appear at the singular point u = u0.

The particles will arrange in some N = 2 supermul-tiplet and will be labeled by some quantum numbers(nm, ne). If the massless particle is purely electric, theBogomol’nyi bound implies a(u0) = 0. It would meanthat the W-bosons become massless at u0 and the wholeSU(2) gauge symmetry is restored there. It would im-ply the existence of a non-Abelian infrared fixed pointwith 〈trφ2〉 6= 0. By conformal invariance, the scaling di-mension of the operator trφ2 at this infrared fixed pointwould have to be zero, i.e., it would have to be the iden-tity operator. It is not possible since trφ2 is odd under aglobal symmetry.

Then, the particles that become massless at the sin-gular point u0 are arranged in an N = 2 supermultiplet

28

of spin ≤ 1/2. The possibilities are severely restrictedby the structure of N = 2 supersymmetry: the multi-plet must be an hypermultiplet that saturates the BPSbound. As we have derived that we should have a 6= 0for all the points of the moduli space, the singular BPSstate must have a non-zero magnetic charge.

Near its associated singularity, the light N = 2 hyper-multiplet is a relevant degree of freedom to be consideredin the low energy Lagrangian. The coupling to the mass-less photon of the unbroken U(1) gauge symmetry has tobe local. Therefore, we apply a duality transformationto describe the relevant degree of freedom (nm, ne) as apurely electric state (0, 1),

(0, 1) = (nm, ne)N−1 , (XXI.1)

with N the appropiate SL(2,Z) transformation. Thedual variables are the good local variables near the u0

singularity. It implies that the monodromy matrix mustleave invariant the singular state (nm, ne). This con-straint plus the U(1) β-function give the monodromy ma-trix

M(nm, ne) =

(1 + 2nmne 2n2

e

−2n2m 1 − 2nenm

). (XXI.2)

In fact, in terms of the local variables,

(a′Da′

)= N

(aDa

), (XXI.3)

the monodromy matrix is just T 2. This result can beunderstood as follows: The renormalizable part of thelow energy Lagrangian is just N = 2 QED with one lighthypermultiplet with mass

√2|a′| =

√2|nmaD + nea|. It

has a trivial infrared fixed point, and the theory is weaklycoupled at large distances. Perturbation theory gives

τ ′ ≃ − i

πlna′ . (XXI.4)

On the other hand, by the monodormy invariance of a′,we have a′(u) ≃ c0(u − u0), this gives the monodromymatrix T 2: τ ′ → τ ′ + 2.

With all the monodromies taken in the counter clock-wise direction, and the monodromy base point chosen inthe negative imaginary part of the complex u plane, wehave the topological constraint

M−Λ2MΛ2 = M∞ . (XXI.5)

If we use the expression (XXI.2) for the monodromiesM±Λ2 and that M∞ = PT−2, (XXI.5) implies that themagnetic charge of the singular states must be ±1. Then,they exist semi-classically and are continuousy connectedwith the weak coupling region. Moreover, if the state(1, ne) becomes massless at u = Λ2, then (XXI.5) givesthe massless state (1, ne−1) at u = −Λ2. It is consistentwith the action of the spontaneously broken symmetryu→ −u, since by the expression of τ(u) in (XVIII.5) we

have that θeff (−Λ2) = 2πRe(τ(−Λ2)) = 2π, and by theWitten effect gives the same physical electric charge tothe massless states at u = ±Λ2.

Seiberg and Witten took the simplest solution: apurely magnetic monopole (1, 0) ††† becomes massless atu = Λ2. With our chosen monodromy base point, thestate with quantum numbers (1,−1) has vanishing massat u = −Λ2.

XXII. THE SEIBERG-WITTEN SOLUTION.

A. The inputs.

After this long preparation, we can present the solu-tion of the model. The moduli space is the compactifiedu-plane punctured at u = Λ2,−Λ2,∞. These singularpoints generate the monodromies:

MΛ2 =

(1 0−2 1

),

M−Λ2 =

(−1 2−2 3

),

M∞ =

(−1 20 −1

), (XXII.1)

which act on the holomorphic function τ(u) by the cor-responding modular transformations. Physically, thefunction τ(u) is the effective coupling at the vacuum uand its asymptotic behavior near the punctured pointsu = Λ2,−Λ2,∞, is known.

B. The geometrical picture.

A torus is a two dimensional compact Riemann surfaceof genus one. Topologically it can be described by a twodimensional lattice with complex periods ω and ωD. Theconstruction is the following: a point z in the complexplane is identifyed with the points z+ω and z+ωD (withthe convention Im(ωD/ω) > 0), to get the topology of atorus. Then, the SL(2,Z) transformations

(ωDω

)→M

(ωDω

)(XXII.2)

leave invariant the torus. If we rescale the lattice with1/ω, the torus is characterized just by the modulus

τ ≡ ωDω,

†††Observe that by Witten effect, the shift θ → θ + 2πntransforms (1, 0) → (1, n). There is a complete democracybetween the semi-classical stable dyons.

29

up to SL(2,Z) transformations,

τ ∼ ατ + β

γτ + δ.

Algebraically the torus can be described by a complexelliptic curve

y2 = 4(x− e1)(x − e2)(x− e3) . (XXII.3)

The toric structure arises because of the two Riemmansheets in the x plane joined through the two branch cutsgoing from e1 to e2 and e3 to infinity (see fig. 2).

x-plane

e3e2e1

α

e1 e2

βe3

β

α

e1

e2FIG. 2. The elliptic curve (XXII.3) gives the topology of a

torus.

The lattice periods are obtained by integrating theAbelian differential of first kind dx/y along the two ho-mologically non-trivial one-cycles α and β, with intersec-tion number β · α = 1,

ωD =

∮

β

dx

y,

ω =

∮

α

dx

y. (XXII.4)

They have the property that Imτ > 0.

C. The Physical connection with N = 2 superYang-Mills.

The breakthough of Seiberg and Witten for the solu-tion of the model was the identification of the complexeffective coupling τ(u) at a given vacuum u with the mod-ulus of a u-dependent torus. At any point u of the modulispace, they associated an elliptic curve

y2 = 4

3∏

i=1

(x− ei(u)) , (XXII.5)

with its lattice periods given by (XXII.4).The identification of the physical coupling τ(u) =

∂aD/∂a with the modulus τu = ωD(u)/ω(u) of the el-liptic curve (XXII.5),

τ(u) =∂aD/∂u

∂a/∂u=

∮β dx/y∮α dx/y

= τu , (XXII.6)

leads to the formulae:

aD =

∮

β

λ(u) , (XXII.7)

a =

∮

α

λ(u) , (XXII.8)

where λ(u) is an Abelian differential with the propertythat

∂λ

∂u= f(u)

dx

y+ dg . (XXII.9)

Then, the solution of the problem is reduced to findingthe family of elliptic curves (XXII.5) and the holomorphicfunction f(u). The conditions at the begining of thissection fix a unique solution. The family of elliptic curvesis determined by the monodromy group generated by themonodromy matrices. The matrices (XXII.1) generatethe group Γ(2), the subgroup of SL(2,Z) consisting ofmatrices congruent to the identity modulo 2. It gives theelliptic curves

y2 = (x2 − Λ4)(x− u) . (XXII.10)

Finally, the function f(u) is determined by the asymp-totic behavior of (aD, a) at the singular points. The an-

swer is f = −√

2/4π.

XXIII. BREAKING N = 2 TO N = 1.MONOPOLE CONDENSATION AND

CONFINEMENT.

In this section we will exhibit an explicit realizationof the confinement mechanism envisaged by Mandelstam[45] and ’t Hooft’s through the condensation of lightmonopoles.

In the N = 2 model, we have found points in the mod-uli space where the relevant light degrees of freedom aremagnetic particles. Since we have the exact solution ofthe low energy N = 2 model, it would be nice to answerin which phase the dynamics of the model, or controlabledeformations of it, locates the vacuum.

For the N = 2 model we already know from sectionXVIII that N = 2 supersymmetry does not allow thegeneration of a superpotential just for the N = 1 chiralmultiplet of the N = 2 vector multiplet. It means thatthe theory is always in an Abelian Coulomb phase. Theexact solution of the model allowed us to know whichare all the instanton corrections to the low energy La-grangian. Remarkably enough, the instanton series ad-mits a resumation in terms of magnetic variables.

To go out of the Coulomb branch, we need a super-potential for the chiral superfield Φ. In [13] an explicit

30

mass term for the chiral superfield was added in the bareLagrangian,

Wtree = m trΦ2 . (XXIII.1)

It breaks N = 2 to N = 1 supersymmetry. Atlow energy, we will have an effective superpotential

W(m,M, M,AD). Once again, holomorphy of the su-perpotential and selection rules from the symmeries willfix the exact form of W . In terms of N = 1 superspace,only the subgroup U(1)J ⊂ SU(2)R is manifestly a sym-metry. It is a non-anomalous R-symmetry (rotates thecomplex phases of θ(I), I = 1, 2, in opposite directions.).The corresponding charge of Φ is zero. As superpoten-tials should have charge two, from (XXIII.1) we derivethat the parameter m 6= 0 breaks the U(1)J symmetryby two units. On the other hand, the N = 1 chiral su-

perfields M and M are in an N = 2 hypermultiplet andtherefore, both have charge one. Imposing that W is a

regular function at m = MM = 0, we find that it is of

the form W = mf1(AD) + MMf2(AD). For m → 0,the effective superpotential flows to the tree level super-

potential (XXIII.1) plus the term√

2ADMM . As thefunctions f1 and f2 are independent of m, we obtain theexact result

W =√

2ADMM +mU(AD) . (XXIII.2)

We found what we were looking for: an exact effectivesuperpotential with a term which depends only of theN = 1 chiral composite operator U . It presumely willremove the flat direction. The N = 2 to N = 1 breakingmakes no loger valid the hiden N = 2 holomorphy inthe Kahler potential K(A,A). But as long as there isan unbroken supersymmetry, the vacuum configurationcorresponds to the solution of the equations

dW = 0 , (XXIII.3)

D = |M |2 − |M |2 = 0 . (XXIII.4)

From the exact solution we know that du/daD 6= 0 ataD = 0. Thus (up to gauge transformations)

M = M =(−mu′(0)/

√2)1/2

,

aD = 0 . (XXIII.5)

Expanding around this vacuum we find:i) There is a mass gap of the order (mΛ)1/2.ii) The objects that condense are magnetic monopoles.

There are electric flux tubes with a non-zero string ten-sion of the order of the mass gap, that confines the elec-tric charges of the U(1) gauge group.

The spontaneously broken symmetry u → −u carriesthe theory to the ‘dyon region’, with the local variableaD − a. The perturbing superpotential there, mU(aD −a), also produces the condensation of the ‘dyon’ withphysical electric charge zero at the point aD − a = 0.Then, we have two physically equivalent vacua, relatedby an spontaneously broken symmetry, in agreement withthe Witten index of N = 1 SU(2) gauge theory.

XXIV. BREAKING N = 2 TO N = 0.

When the N = 2 theory is broken to the N = 1 theorythrough the decoupling of the chiral superfield Φ in theadjoint, we have seen that the mechanism of confinementtakes place because of the condensation of a magneticmonopole. The natural question is if this results can beextended to non supersymmetric gauge theories.

The N = 1, 2 results were based on the use of holo-morphy; the question is whether the properties connectedwith holomorphy can be extended to theN = 0 case. Theanswer is positive provided supersymmetry is broken viasoft breaking terms.

The method is to promote some couplings in the su-persymmetric Lagrangian to the quality of frozen super-fields, called spurion superfields. We could think theycorrespond to some heavy degrees of freedom which atlow energies have been decoupled. Their trace is onlythrough their vacuum expectation values appearing inthe Lagrangian and are parametrized by the spurion su-perfields [46].

In the N = 2 theory we will promote some couplings tothe status of spurion superfields. The property of holo-morphy in the prepotential will be secured if the intro-duced spurions are N = 2 vector superfields [14,15] ‡‡‡.

In the bare Lagrangian of the N = 2 SU(2) gauge the-ory (XVII.3), there is only one parameter: τ0. TheN = 2softly broken theory is obtained by the bare prepotential

F0 =1

πSAaAa , (XXIV.1)

where S is an dimensionless N = 2 vector multipletwhose scalar component gives the bare coupling constant,s = π

2 τ0. The factor of proporcionality is related withthe one loop coefficient of the beta function, such thatΛ = µ0exp(is). Inspired by String Theory, we call Sthe dilaton spurion. The source of soft breaking comesfrom the non vanishing auxiliary fields, F0 and D0, in thedilaton spurion S.

The tree level mass terms arising from the softly bro-ken bare Lagrangian (XXIV.1) are the following: the W-bosons get a mass term by the usual Higgs mechanism,with the mass square equal to 2|a|2; the photon of the un-broken U(1) remains massless; the gauginos get a masssquare M2

1/2 = (|F0|2 + D20/2)(4Ims)−1; all the scalar

components, except the real part of φ3 which do not havea bare mass term, get a square mass M2

0 = 4M21/2.

At low energy, i.e., at scales of the order |u|1/2 ∼ Λ, theWilsonian effective Lagrangian up to two derivatives andfour fermions terms is given by the effective prepotentialF(a,Λ) found in the N = 2 model, but with the differ-ence that the bare coupling constant is replaced by the

‡‡‡Soft breaking of N = 1 SQCD has been studied in [47].

31

dilaton spurion, i.e., Λ → µ0exp(iS). Then, the prepo-tential depends on two vector multiplets and the effectiveLagrangian becomes

L =1

4πIm

(∫d4θ

∂F∂Ai

Ai+

∫d2θ

1

2

∂2F∂Ai∂Aj

W iW j

)

+ LHM . (XXIV.2)

with Ai = (S,A) and LHM the N = 2 Lagrangian thatincludes the monopole hypermultiplet. Observe that thedilaton spurion do not enter in the Lagrangian of thehypermultiplets, in agreement with the N = 2 non-renormalization theorem of [27]. The low energy cou-plings are determined by the 2 × 2 matrix

τij(a, s) =∂2F∂ai∂aj

. (XXIV.3)

The supersymmetry breaking generates a non-trivial ef-fective potential for the scalar fields,

Veff =

(b00 −

b201b11

)(|F0|2 +

1

2D2

0

)

+b01b11

[√2(F0mm+ F 0mm) +D0(|m|2 − |m|2)

]

+1

2b11(|m|2 + |m|2)2 + 2|a|2(|m|2 + |m|2) , (XXIV.4)

where we have defined bij = (4π)−1Imτij . m and m arethe scalar components of the chiral superfields M and

M of the monopole hypermultiplet, respectively. Ob-serve that the first line of (XXIV.4) is independent ofthe monopole degrees of freedom. To be sure that suchquantity gives the right amount of energy at any pointof the moduli space, where different local descriptions ofthe physics are necessary, it must be duality invariant.This is the case for any SL(2,Z) transformation.

The auxiliary fields of the dilaton spurion are in theadjoint representation of the group SU(2)R and haveU(1)R charge two. We can consider the situation ofD0 = 0, F0 = f0 > 0 without any loss of generality,since it is related with the case of D0 6= 0 and complexF0 just by the appropiate SU(2)R rotation.

We have to be careful with the validity of our approx-imations. Because of supersymmetry, the expansion inderivatives is linked with the expansion in fermions andthe expansion in auxiliary fields. The exact solution ofSeiberg and Witten is only for the first terms in thederivative expansion of the effective Lagrangian, in par-ticular up to two derivatives. At the level of the softlybroken effective Lagrangian, the exact solution of Seibergand Witten only gives us the terms at most quadratic inthe supersymmetry breaking parameter f0. The expan-sion is performed in the dimensionless parameter f0/Λ.Our ignorance on the higher derivative terms of the effec-tive Lagrangian is traslated into our ignorance the termsof O((f0/Λ)4). Hence our results are reliable for smallvalues of f0/Λ, and this is far from the supersymmetrydecoupling limit f0/Λ → ∞.

But for moderate values of the supersymmetry break-ing parameter, the effective Lagrangian (XXIV.2) givesthe large distance physics of a non-supersymmetric gaugetheory at strong coupling. If we minimize the effectivepotential (XXIV.4) with respect to the monopoles, weobtain the energy of the vacuum u

Veff (u) =

(b00(u) −

b201(u)

b11(u)

)|F0|2

− 2

b11(u)ρ4(u) , (XXIV.5)

where ρ(u) is a positive function that gives the monopolecondensate at u

|m|2 = |m|2 = ρ2(u) =|b01|f0√

2− b11|a|2 > 0 (XXIV.6)

or m = m = ρ(u) = 0 if |b01|f0 <√

2b11|a|2.

0

0.5

1

1.5

2 -1

-0.5

0

0.5

1

0

0.002

0.004

0.006

0.008

0

0.5

1

1.5

2

FIG. 3. The monopole condensate ρ2, at the monopole re-gion u ∼ Λ2, for f0 = Λ/10.

Notice that b11 diverges logarithmically at the singu-larities u = ±Λ2, but the corresponding local variable avanishes linearly at u = ±Λ2. It implies that b11|a|2 → 0for u → ±Λ2. It can be shown that the Seiberg-Wittensolution gives b01 ∼ Λ/8π for u ∼ Λ. It means thatthe monopole condenses at the monopole region (seefig. 3), since from the expression of the effective potential(XXIV.5), such condensation is energetically favoured. Ifwe look at the dyon region, we find that b01 → 0 foru → −Λ2. Numerically, there is a very small dyon con-densate without any associated minimum in the effectivepotential in that region. On the other hand, there isa clear absolute minimum in the monopole region (seefig. 4). The different behaviors of the broken theory un-der the transformation u → −u is an expected result ifwe take into account that f0 6= 0 breaks explicitly theU(1)R symmetry.

32

0.6

0.8

1

1.2

1.4-0.4

-0.2

0

0.2

0.4

-0.001

-0.0005

0

0.6

0.8

1

1.2

1.4

FIG. 4. The effective potential Veff (u) (XXIV.5), at themonopole region u ∼ Λ2, for f0 = Λ/10.

The softly broken theory selects a unique minimum atthe monopole region, with a non vanishing expectationvalue for the monopole. The theory confines and has amass gap or order (f0Λ)1/2.

XXV. STRING THEORY IN PERTURBATIONTHEORY.

String Theory is a multifaceted subject. In the sixtiesstrings were first introduced to model the dynamics ofhadron dynamics. In section VII we described the con-fining phase as the dual Higgs phase, where magneticdegrees of freedom condense. The topology of the gaugegroup allows the existence of electric vortex tubes, end-ing on quark-antiquark bound states. The transverse sizeof the electric tubes is of the order of the compton wavelength of the ‘massive’ W-bosons. At large distances,these electric tubes can be considered as open stringswith a quark and an anti-quark at their end points. Thisis the QCD string, with an string tension of the or-der of the characteristic length square of the hadrons,α′ ∼ (1GeV)−2.

But the major interest in String Theory comes frombeing a good candidate for quantum gravity [48]. Themacroscopic gravitational force includes an intrinsic con-stant, GN , with dimensions of length square

GN = l2p = (1.6 × 10−33cm)2 . (XXV.1)

In a physical process with an energy scale E for the fun-damental constituents of matter, the strength of the grav-itational interaction is given by the dimensionless cou-pling GNE

2 to the graviton. This interaction can beneglected when the graviton probes length scales muchlarger than the Planck’s size, GNE

2 ≪ 1. The interac-tion is also non-renormalizable. From the point of view

of Quantum Field Theory, it corresponds to an effectivelow energy interaction, with lp the natural length scaleat which the effects of quantum gravity become impor-tant. The natural suspicion is that there is new physics atsuch short distances, which smears out the interaction.The idea of String Theory is to replace the point par-ticle description of the interactions by one-dimensionalobjects, strings with size of the order of the Planck’slength lp ∼ 10−33cm (see fig. 5). Such simple changehas profound consequences on the physical behavior ofthe theory, as we will briefly review below. It is still notclear whether the stringy solution to quantum gravityshould work. Because Planck’s length scale is so small,up to now String Theory is only constructed from internalconsistency. But it is at the moment the best candidatewe have. Let us quickly review some of the major impli-cations of String Theory, derived already at perturbativelevel.

graviton

FIG. 5. The point particle graviton interchange is replacedby the smeared string interaction.

The first important consequence of String Theory isthe existence of vibrating modes of the string. Theycorrespond to the physical particle spectrum. For phe-nomenology the relevant part comes from the masslessmodes, since the massive modes are excited at energiesof the order of the Planck’s mass l−1

p . At low energies allthe massive modes decouple and we end with an effectiveQuantum Field Theory for the massless modes. In themassless spectrum of the closed string, there is a parti-cle of spin two. It is the graviton. Then String Theoryincludes gravity. If we know how to make a consistentand phenomenologically satisfactory quantum theory ofstrings, we have quantized gravity.

Up to now, String Theory is only well understood atthe perturbative level. The field theory diagrams are re-placed by two dimensional Riemann surfaces, with theloop expansion being performed by an expansion in thegenus of the surfaces. It is a formulation of first quanti-zation, where the path integral is weighed by the area ofthe Riemann surface and the external states are includedby the insertion of the appropiate vertex operators (seefig. 6). The perturbative string coupling constant is de-termined by the vacuum expectation value of a masslessreal scalar field, called the dilaton, through the relationgs = exp〈s〉. The thickening of Feynman diagrams into‘surface’ diagrams improves considerably the ultravioletbehavior of the theory. String Theory is ultraviolet finite.

33

+ + + . . . .

FIG. 6. The preturbative loop expansion in String Theoryis equivalent to expand in the number of genus of the Riemannsurfaces.

The third important consequence is the introductionof supersymmetry. For the bosonic string, the lowestvibrating mode correponds to a tachyon. It indicatesthat we are performing perturbation theory arround anunestable minimum. Supersymmetry gives a very eco-nomical solution to this problem. In a supersymmetrictheory the hamiltonian operator is positive semi-definiteand the ground state has always zero energy. It is alsovery appealing from the point of view of the cosmologi-cal constant problem. Furthermore, supersymmetry alsointroduces fermionic degrees of freedom in the physicalspectrum. If nature really chooses to be supersymmet-ric at sort distances, the big question is: How is su-persymmetry dynamically broken? The satisfactory an-swer must include the observed low energy phenomena ofthe standard model and the vanishing of the cosmologi-cal constant. As a last comment on supersymmetry wewill say that the Green-Schwarz formulation of the su-perstring action demands invariance under a world-sheetlocal fermionic symmetry, called κ-symmetry. It is onlypossible to construct κ-symmetric world-sheet actions ifthe number of spacetime symmetries is N ≤ 2 (in tenspacetime dimensions).

The fourth important consequence is the prediction onthe number of dimensions of the target space where theperturbative string propagates. Lorentz invariance onthe target space or conformal invariance on the world-sheet fixes the number of spacetime dimensions (twenty-six for bosonic strings and ten for superstrings). As ourlow energy world is four dimensional, String Theory in-corporates the Kaluza-Klein idea in a natural way. Butagain the one-dimensional nature of the string gives aquite different behavior of String Theory with respect tofield theory. The dimensional reduction of a field the-ory in D spacetime dimensions is another field theory inD−1 dimensions. The effect of a non-zero finite radius Rfor the compactified dimension is just a tower of Kaluza-Klein states with masses n/R. But in String Theory, thestring can wind m times around the compact dimension.This process gives a contribution to the momentum ofthe string proportional to the compact radius, mR/α′.These quantum states become light for R → 0. The di-mensional reduction of a String Theory in D dimensionsis another String Theory in D dimensions. This is Tduality [49].

The fifth important consequence comes from the can-cellation of spacetime anomalies (gauge, gravitationaland mixed anomalies). It gives only the following fiveanomaly-free superstring theories in ten spacetime di-

mensions.

A. The type IIA and type IIB string theories.

A type II string theory is constructed from closedsuperstrings with N = 2 spacetime supersymmetries.The spectrum is obtained as a tensor product of a left-and right-moving world-sheet sectors of the closed string.Working in the light-cone gauge, the massless states ofeach sector are in the representation 8v ⊕8± of the littlegroup SO(8). The representations 8v and 8± are the vec-tor representation and the irreducible chiral spinor rep-resentations of SO(8), respectively.

The type IIA string theory corresponds to the choiceof opposite chiralities for the spinorial representations inthe left- and right-moving sectors,

Type IIA : (8v ⊕ 8+) ⊗ (8v ⊕ 8−) . (XXV.2)

The bosonic massless spectrum is divided between theNS-NS fields:

8v ⊗ 8v = 1⊕ 28⊕ 35 , (XXV.3)

which corresponds to the dilaton s, the antisymmetrictensor Bµν and the gravitation field gµν , respectively,and the R-R fields:

8+ ⊗ 8− = 8v ⊕ 56, (XXV.4)

which correspond to the light-cone degrees of freedom ofthe antisymmetric tensors Aµ and Aµνρ, respectively. Asthe chiral spinors have opposite chiralities, in the vertexoperators of the R-R fields only even forms appear, F2

and F4. The physical state conditions on the masslessstates give the following equations on these even forms:

dF = 0 d ⋆ F = 0 , (XXV.5)

with ⋆F the Poincare dual (10 − n)-form of the n-formFn. These are the Bianchi identity and the equation ofmotion for a field strength. Their relation with the R-Rfields is then Fn = dAn−1. The Abelian field strengthsFn are gauge invariant, and since these are the fields thatappear in the vertex operators, the fundamental stringsdo not carry RR charges.

The fermionic massless spectrum is given by the NS−R and R−NS fields:

8v ⊗ 8− = 8+ ⊕ 56− ,

8+ ⊗ 8v = 8− ⊕ 56+ . (XXV.6)

The 8± states are the two dilatini. The 56± states are thetwo gravitini, with a spinor and a vector index. Observethat the fermions have opposite chiralities, which preventthe type IIA theory from gravitational anomalies.

The Type IIB String Theory corresponds to the choiceof the same chirality for the spinor representations of theleft- and right-moving sector,

34

Type IIB : (8v ⊕ 8+) ⊗ (8v ⊕ 8+) . (XXV.7)

The NS-NS fields are the same as for the type IIA string.The difference comes from the R-R fields:

8+ ⊗ 8+ = 1 + ⊕28⊕ 35+ . (XXV.8)

They correspond, respectively, to the forms A0, A2 andA4 (self-dual).

For the massless fermions there are two dilatini andtwo gravitini, but now all of them have the same chiral-ity. In spite of it, the theory does not have gravitationalanomalies [50].

Under spacetime compactifications, the type IIA andthe type IIB string theories are unified by the T -dualitysymmetry. It is an exact symmetry of the theory alreadyat the perturbative level and maps a type IIA string witha compact dimension of radius R to a type IIB string withradius α′/R.

B. The Type I string theory.

It is constructed from unoriented open and closed su-perstrings, leading only N = 1 spacetime supersymme-try. The massless states are:

Open : 8v ⊗ 8+ (XXV.9)

Closed sym. : [(8v ⊕ 8+) ⊗ (8v ⊕ 8+)]sym =

= [1⊕ 28⊕ 35]bosonic ⊕ [8− ⊕ 56−]fermionic . (XXV.10)

The massless sector of the spectrum that comes from theunoriented open superstring (XXV.9) gives N = 1 su-per Yang-Mills theory, with a gauge group SO(Nc) orUSp(Nc) introduced by Chan-Paton factors at the endsof the open superstring. The sector coming from theunoriented closed string (XXV.10) gives N = 1 super-gravity. Cancellation of spacetime anomalies restricts thegauge group to SO(32).

C. The SO(32) and E8 × E8 heterotic strings.

The heterotic string is constructed from a right-moving closed superstring and a left-moving closedbosonic string. Conformal anomaly cancellation de-mands twenty-six bosonic target space coordinates in theleft-moving sector. The additional sixteen left-movingcoordinates XI

L, I = 1, ..., 16, are compactified on a T 16

torus, defined by a sixteen-dimensional lattice, Λ16, withsome basis vectors eIi , i = 1, ..., 16. The left-moving

momenta pIL live on the dual lattice Λ16. The mass oper-

ator gives an even lattice (∑16

I=1 eIi eIi = 2 for any i). The

modular invariance of the one-loop diagrams restricts the

lattice to be self-dual (Λ16 = Λ16). There are only twoeven self-dual sixteen-dimensional lattices. They corre-spond to the root lattices of the Lie groups SO(32)/Z2

and E8 × E8.

For the physical massless states, the supersymmetricright-moving sector gives the factor 8v ⊗ 8+, which to-gether with the lattice points of length squared two ofthe left-moving sector, give an N = 1 vector multiplet inthe adjoint representation of the gauge group SO(32) orE8 × E8.

There is also a T -duality symmetry relating the twoheterotic strings.

XXVI. D-BRANES.

Perturbation theory is not the whole history. In thefield theory sections we have learned how much the non-perturbative effects could change the perturbative pic-ture of a theory. In particular, there are nonperturbativestable field configurations (solitons) that can become therelevant degrees of freedom in some regime. In that situa-tion it is convenient to perform a duality transformationto have an effective description of the theory in termsof these solitonic degrees of freedom as the fundamentalobjects.

What about the nonperturbative effects in String The-ory?. Does String Theory incorporate nonperturbativeexcitations (string solitons)?. Are there also strong-weakcoupling duality transformations in String Theory?. Be-fore the role of D-branes in String Theory were appreci-ated, the answers to these three questions were not clear.

For instance, it was known, by the study of large ordersof string perturbation theory, that the nonperturbativeeffects in string theory had to be stronger than in fieldtheory, in the sense of being of the order of exp(−1/gs)instead of order exp(−1/g2

s) [51], but it was not knownwhich were the nature of such nonperturbative effects.

With respect the existence of nonperturbative objects,the unique evidence came form solitonic solutions of thesupergravity equations of motion which are the low en-ergy limits of string theories. These objects were in gen-eral extended membranes in p + 1 dimensions, called p-branes [52].

In relation to the utility of the duality transformationin String Theory, there is strong evidence of some stringdualities [53]. There is for instance the SL(2,Z) self-duality conjecture of the type IIB theory [54]. Under anS-transformation,the string coupling value gs is mappedto the value 1/gs, and the NS-NS field Bµν is mappedto the R-R field Aµν . Then, self-duality of type IIB de-mands the existence of an string with a tension scalingas g−1

s and non-zero RR charge.

A. Dirichlet boundary conditions.

In open string theory, it is possible to impose two dif-ferent boundary conditions at the ends of the open string:

Neuman : ∂⊥Xµ = 0 . (XXVI.1)

Dirichlet : ∂tXµ = 0 . (XXVI.2)

35

An extended topological defect with p+ 1 dimensionsis described by the following boundary conditions on theopen strings:

∂⊥X0,1,···p = ∂tX

p+1,···9 = 0 . (XXVI.3)

We call it a D p-brane (for Dirichlet [55]), an extended(p+1)-dimensional object (located at Xp+1,···9 = const)with the end points of open strings attached to it.

The Dirichlet boundary conditions are not Lorentz in-variant. There is a momentum flux going from the ends ofopen strings to the D-branes to which they are attached.In fact, the quantum fluctuations of the open string end-points in the longitudinal directions of the D-brane liveon the world-volume of the D-brane. The quantum fluc-tuations of the open string endpoints in the transversedirections of the D-brane, makes the D-brane fluctuatelocally. It is a dynamical object, characterized by a ten-sion Tp and a RR charge µp. If µp 6= 0, the world-volumeof a p-brane will couple to the R-R (p+ 1)-form Ap+1.

Far from the D-brane, we have closed superstrings, butthe world-sheet boundaries (XXVI.3) relates the right-moving supercharges to the left-moving ones, and onlya linear combination of both is a good symmetry of thegiven configuration. In presence of the D-brane, half ofthe supersymmetries are broken. The D-brane is a BPSstate. In fact, in [56] it was shown that the D-branetension arises from the disk and therefore that it scalesas g−1

s . This is the same coupling constant dependenceas for BPS solitonic branes carrying RR charges [52].

The Dirichlet boundary condition becomes the Neu-man boundary condition in terms of the T -dual coordi-nates, and vice versa. It implies that if we T -dualizea direction longitudinal to the world- volume of the Dp-brane, it becomes a (p − 1)-brane. Equally, if the T -dualized direction is transverse to the D p-brane, we ob-tain a D (p + 1)-brane. Consider a 9-brane in a typeIIB background. The 9-brane fills the spacetime and theendpoints of the open strings attached to it are free tomove in all the directions. It is a type I theory, withonly N = 1 supersymmetry. Now T -dualize one direc-tion of the target space. We obtain an 8-brane in a typeIIA background. If we proceed further, we obtain thata type IIB background can hold p = 9, 7, 5, 3, 1,−1 p-branes. A D (−1)-brane is a D-instanton, a localizedspacetime point. For a type IIA background we obtainp = 8, 6, 4, 2, 0 p-branes.

B. BPS states with RR charges.

FIG. 7. Two parallel D-branes with the one-loop vacuumfluctuation of an open string attached between them. Bymodular invariance, it also corresponds to a tree level inter-change of a closed string.

To check if really the D-branes are the nonperturba-tive string solitons required by string duality, Polchinskicomputed explicitly the tension and RR charge of a Dp-brane [57]. He first computed the one-loop amplitudeof an open string attached to two parallel D p-branes.The resulting Casimir force between the D-branes waszero, supporting its BPS nature. By modular invariance,it can also be interpreted as the amplitude for the inter-change of a closed string between the D-branes (see fig.7). In the large separation limit, only the massless closedmodes contribute. These are the NS-NS fields (gravitonand dilaton) and the R-R (p + 1) form. On the spacebetween the D-branes these fields follow the low energytype II action (type IIA for p even and type IIB for podd). On the D p-branes, the coupling to the NS-NS andR-R fields is

Sp = Tp

∫dp+1ξ e−s |detGab|1/2 + µp

∫

p−brane

Ap+1 .

(XXVI.4)

From (XXVI.4) we see that the actual D-brane actionincludes a dilaton factor τp = Tp/gs, with gs the couplingconstant of the closed string theory. Comparing the fieldtheory calculation with the contribution of the masslessclosed modes in the string theory computation, one canobtain the values of Tp and µp. The result is [57]

µ2p = 2T 2

p = (4π2α′)3−p . (XXVI.5)

Observe that the R-R charge is really non-zero. In fact,if one checks (the generalization of) the Dirac’s quanti-zation condition for the charge µp and its dual chargeµ(6−p), one obtains that µpµ(6−p) = 2π. They satisfythe minimal quantization condition. It means that theD-branes carry the minimal allowed RR charges.

36

XXVII. SOME FINAL COMMENTS ONNONPERTURBATIVE STRING THEORY.

A. D-instantons and S-duality.

The answers to the three questions at the beginningof the previous section can now be more concrete, sincesome nonperturbative objects in String Theory has beenidentified: the D-branes.

Consider a D p-brane wrapped around a non-trivial(p + 1)-cycle. This configuration is topologically stable.Its action is TpVp+1/gs, with Vp+1 the volume of the non-trivial (p + 1) cycle. It contributes in amplitudes withfactors e−TpVp+1/gs , a generalized instanton effect. Nowwe understand why the nonperturbative effects in StringTheory are stronger than in field theory, it is related tothe peculiar nature of the string solitons.

The D-branes also give the necessary ingredient for theSL(2,Z) self-duality of the type IIB string theory. Thistheory allowsD 1-branes, with a mass τ1 ∼ (2πα′gs)−1 inthe string metric and non-zero RR charge. Also, one cansee that on the D 1-brane there are the same fluctuationsof a fundamental IIB string [58]. Then, it is the requiredobject for the S-duality transformation of the type IIBstring. In fact, at strong coupling the D 1-string becomeslight and it is natural to formulate the type IIB theoryin terms of weakly coupled D 1-branes.

There is another S-duality relation in String Theory.Observe that the type I theory and the SO(32) heterotictheory have the same low energy limit. It could be thatthey correspond to the same theory but for different val-ues of the string coupling constant. Again D-branes helpto make this picture clearer. Consider a D 1-brane in atype I background with open strings attached to it, butalso with open strings with one end point attached to a9-brane. We call them 1 − 9 strings. The 9-brane fillsthe spacetime, and the 1 − 9 strings, having one Chan-Paton index, are vectors of SO(32). One can see thatthe world-sheet theory of the D 1-brane is precisely thatof the SO(32) heterotic string [59]. Having a tensionthat scales as g−1

s , one can argue that this D heteroticstring sets the lightest scale in the theory when gs ≫ 1.The strong coupling behavior of the type I string can bemodeled by the weak coupling behavior of the heteroticstring.

B. An eleventh dimension.

Type IIA allows the existence of 0-branes that cou-ple to the R-R one-form A1. The 0-brane mass isτ0 ∼ (α′)−1/2/gs in the string metric. At strong cou-pling in the type IIA theory, gs ≫ 1, this mass is thelightest scale of the theory. In fact, n 0-branes can forma BPS bound state with mass nτ0. This tower of statesbecoming a continuum of light states at strong coupling

is characteristic of the appearance of an additional di-mension. Type IIA theory at strong coupling feels aneleventh dimension of some size 2πR, with the 0-branesplaying the role of the Kaluza-Klein states [60].

If we compactify 11D supergravity [61] on a circle ofradius R and compare its action with the 10D type IIAsupergravity action, we obtain the relation

R ∼ g2/3s . (XXVII.1)

This eleventh dimension is invisible in perturbationtheory, where we perform an expansion near gs = 0.

This has been a lightning review of some aspects of du-ality in String Theory. We hope it will serve to whet theappetite of the reader and encourage her/him to learnmore about the subject and to eventually contribute tosome of the outstanding open problems. More informa-tion can be found from the references [62].

AcknowledgmentsWe have benefited from valuable conversations with

many colleages. We would like to thank in particularE. Alvarez, J.M. F. Barbon, J. Distler, D. Espriu, C.Gomez, J. Gomis, K. Kounnas, J. Labastida, W. Lerche,M. Marino, J.M. Pons and E. Verlinde for discussions.F. Z. would like to thank the Theory Division at CERNfor its hospitality. The work of F. Z. is supported by afellowship from Ministerio de Educacion y Ciencia.

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